Disclaimer: It is a very philosophical question (more so than a worldbuilding one), so this is going to be a very long wall of text. It might be hard to read because I do not agree with suggested criteria of godhood.
Some thoughts on suggested criteria
If I understand correctly, according to your constraints, a god will be considered new only if all of these are true:
- this deity is not a historic human;
- their religion is not an appropriation of an earlier cult or religion and their gods;
- their religion is organised;
- this god is worshipped by a cultural elite;
- this god does not belong to an already existing 'type'.
It seems obvious that the Christian god is not new since it fails criteria #1, #2, and #5:
- the Christians regardless of denomination and their views on the divinity of Christ accept his historicity (and it is the only thing that matters in this case);
- Christianity is an Abrahamic religion built upon Judaism and its sacred texts and reusing its god and mythology;
- the Christian god is the same type of 'moral' god1 as the Judaic god.
However, if we look closer the god of the New Testament is almost the opposite of the god of the Old Testament. The old god was very jealous and selective. Almost all covenants were made exclusively with the Chosen people (the Noahic covenant is the only exception) and the rest were essentially damned. And the god would not hesitate to punish his followers when they disobey or forget about the covenant. He is even more brutal when it comes to those he did not choose. He destroys their lands and cities without much of a thought.
In many ways, the Judaic god resembles a male warrior god from patriarchal societies. He is a strict and not easily forgiving father who likes to test his children a lot. And the same time he leads them to battle and grants them many victories, spoils, and trophy wives.
The Christian god has a very different attitude. It is a motherly figure. The New Covenant is potentially much more inclusive (although, it is open for debate who qualifies for membership) and substantially easier to follow (one only needs to believe, there are no elaborate rituals). The god offers unconditional love and salvation.
It is also important to mention that Christianity introduced a concept of Trinity — the Father, the Son, and the Holy Spirit. Christian theologists have to use very complex arguments to resolve the problem of three personas of one single god within monotheism. Judaism is not only not accepting of this idea but considers it heretical. The Judaic god is always absolute and indivisible.
Historically the Judaic god was the god of conquerors and kings while the Christian god was the god of slaves. And this is reflected in theology and their attitudes.
With this said, while the Christian god was derived from the Judaic god, the Christianity still uses Judaic texts, and there are quite some similarities between the gods of these two religions, the differences are substantial enough to say that Christianity created a new god.
To further address your criteria.
I do think that in our modern society the surest way to create a new god is to convince the general public of the divinity of a living breathing human being or them being a proxy of the divine. Seeing is believing. Thus, the public needs to see miracles. And I am not talking about smoke and mirrors. I mean real miracles2. People cured of lethal diseases. Water changed into wine. This kind of things.
I believe it is almost impossible to avoid appropriation, borrowing, or any other influence from already existing religions if there is no complete isolation. For example, the Roman Catholic Church has slightly different saints depending on location. And those saints frequently possess the features of pagan gods Christianity replaced in this area. Christianity is a monotheistic religion, but people ask saints for favours exactly the same way they would ask minor gods of a pagan pantheon. The old ways are slow to die. And any new religion will incorporate at least some elements of the pre-existing ones. Unless it is created artificially in a completely sterile cultural environment, i.e. people's memories will be completely wiped out.
Organised religion with its priests, temples, and grandeur, of course, adds political weight. However, I am not sure that it is a necessary condition for a true belief in a god. In some sense, Virgin Mary is a goddess without an organised religion devoted personally to her. Yet, she has a huge following, especially in Central America. Pre-historic religions, Native American religions, branches of Hinduism, etc. also had gods but their belief system was not institutionalised.
I am also not sure that if a god is not worshipped by a cultural elite automatically makes their religion a cult. We can somewhat argue that Christianity was a cult in the Roman Empire prior to Constantine because the elites mostly practised paganism. Christianity was most popular among low classes (it is more complicated than that, of course). And once Christianisation of the Roman Empire happened the Christianity finally graduated. But what about Christianisation of other countries? Vikings and Russian princes adopted Christianity for political reasons. The general population continued to worship pagan gods for at least a couple more centuries. If any of those pagan pantheons acquired a new god would it be just a cult?
I think a more satisfactory criterion will be a number of followers than simply worshipping by cultural elites. Cultural elites are also prone to creating cults. In today's USA, for example, it is a cult of money. Everything is measured in money. In fact, you can say that money is the new god.
Please see Note 1 for my thoughts on 'types' of gods. I do not think that you can really come up with something radically new. You can have a god responsible for the material world or a god responsible for morals. Monotheistic religions usually offer the 2-in-1 package with more emphasis on morals. Perhaps once we solve the consciousness riddle we will be able to come up with another type of god.
The modern gods
While we are losing faith in traditional gods and abandoning churches and temples we are not becoming rational beings. We just replace old gods with the new ones and disguise them as non-religious things.
A curious case of the US Constitution or the goddess of Democracy
Disclaimer: This is not to start a flame war.
The US Constitution enjoys almost the same sacred status as the Bible in the USA. It cannot be changed. It is THE divine truth. Everybody ought to accept it. It has to be worshipped and it is worshipped. Even a thought of changing it (because it was written 250 years ago when the world was different, the slavery was a thing, and modern technology did not exist even in people's imagination) is a blasphemy.
Notes on the Constitution amendments (to address some comments).
There is a grand total of 27 (twenty-seven) fully ratified amendments to the US Constitution since March 4, 1789 (the Constitution ratification date). The first 10 are known as the Bill of Rights. They cover rights and freedoms (that somehow went missing from the final text of the Constitution) and were put to vote as one single package. The ratification process was complete in December 1791.
Only 17 amendments were added in the following 227 years. Moreover, after 1971, when the last of the Civil Rights era amendments was added, there was only one amendment and it was originally proposed in 1789. It took over 200 years to finally add it to the US constitution which happened in 1992. A funny detail: The final ratification of the Amendment XXVII was a personal project of just one man.
We could blame the Article V of the US Constitution for the lack of amendments. The required procedure is indeed incredibly demanding. However, this might be not the only reason.
According to the US Senate,'approximately 11,699 measures have been proposed to amend the Constitution from 1789 through January 3, 2017.' If you follow the link you can notice a downward trend in number of proposals starting from the 105th Congress (1997-1998). Although, without more data it is hard to say whether it is statistically valid.
The last time a proposed amendment went to states for ratification was 1978 (it would allow Washington DC to have representatives in Congress). But it failed. The absolute majority of proposals do not even get voted in Congress. They just die in committees and workgroups.
The US Constitution is an exception from normal practice in the USA. State constitutions are amended regularly (although, the procedures of amendment are usually much less demanding):
The Maryland State Constitutions Project reported in 2000 that "there have been almost 150 state constitutions, they have been amended roughly 12,000 times, and the text of the constitutions and their amendments comprises about 15,000 pages of text."
From 2006 through 2014, 683 constitutional amendments were proposed and put before voters, and 482 amendments were approved.
For those who are interested to see how the US Constitution fares in comparison with constitutions of other countries the Comparative Constitutions Project made a really neat Timeline of Constitutions chart which lists new constitutions, amendments, and their dates for many countries in the world, including some historical states.
I think that the Constitution is one of the few modern text so closely resembling religious texts in status. It enjoys the same awe. And it also fuels very heated debates. Moreover, it is brilliantly short and open to interpretation.
I would say that the Constitution is essentially a sacred text of the goddess of Democracy. Democracy here is specifically the US brand. European democracies do not really count as they are tainted by socialism (which acts as a demon or even Satan).
Democracy is irrationally believed to bring happiness and prosperity to everyone regardless of their cultural and historical conditions. Democracy is forced on pretty much all the countries in the world. And if it does not work the people of a country are blamed for not embracing the values of democracy. This is way too similar to religious narratives that gods answer only to the faithful.
The money god
Money is another modern god, especially if one lives in an Americanised society. One's happiness, one's worth, one's social standing, and so on are measured in money. According to the belief, the more money one has the better one is. This is total BS, of course. But faith is irrational. So people continue to chase the elusive god of money.
Technology and science
The previous gods were minor compared to science and technology which almost replaced our beliefs in supernatural beings. Contemporary science and technology are beyond the grasp of ordinary people. Even specialists may not understand how something not related to their field works. It is really close to true miracles.
Science and technology make promises that only gods before could make:
Freedom from disease, eternal youth, immortality, endless exploration, victories on any battlefields, and so on.
Despite the fact that contemporary science is in crisis because a significant part of the experiments cannot be replicated and scandals involving financing and data manipulation, we continue to believe in science. Technology is literally destroying the planet, but we hope that it will be our salvation.
The preference for science and technology is reflected in education as well. The proposals to reduce hours for non-STEM disciplines and to focus on the STEM are quite frequent. Online discussions are full of remarks on 'useless' arts and humanities. International prestige of a country is measured based on its children's performance in maths, science, and reading.
One can argue that recent events in the USA demonstrate anti-science attitudes. I would agree with that with one caveat. New regulations and policies are often pro-technology (and pro-money, of course). However, those might be outdated technologies that current political powers are the most familiar with.
I am sure that if I think long enough I will be able to come up with some other new gods. A god of the eternal youth looks like a good candidate. But it is already a very long answer.
I do not think that modern gods will satisfy your criteria. Modern mindset is very different and blind faith in miracles and divinity is not something that can fit contemporary life easily. Unless divinity can be demonstrated to people, handed to them on a platter. But I still believe that we are not ready to abandon faith and old gods will be replaced by new ones. However, we will rationalise them and disguise them as values and worldviews. The saddest thing is that we will kill others for them exactly the same way we were killing thousands of years ago. Because one thing religious fanatics cannot tolerate is heretics.
1 All gods can be divided into two main categories:
- representations of the environment (forces of nature, elements of landscape, etc, e.g. Shinto gods, Greek and Roman gods);
- representations of moral and ethical standards (e.g. Abrahamic gods).
2 If we are talking about fabricating a new religion, we still need 'real' miracles. Magician tricks will not work. I think there are two approaches to this: 1) technology so advanced that it is not distinguishable from a divine intervention mixed with some psychological manipulation and showing off; 2) move miracles from physical to mental plane, i.e. give intellectual elite a system of values and philosophical views that change their life outlooks.