John Dallman's answer makes the essential point. True language is not restricted in the messages it can send.
In the 1960s a linguistic anthropologist called Charles F. Hockett set out an influential list of "design features" that all human languages share. I am not claiming that Hockett was necessarily correct - he himself revised the original list, and many subsequent observers have pointed out that he did not look at sign languages used by the deaf. His list only deals with Earthly creatures. Nonetheless the list has often been used by scientists and science fiction writers as a basis for thinking about what separates true language from animal language. Many of the items on his list were qualities shared between human and animal communication but he held the following to be markers of true language (quoting from the Wikipedia link):
Displacement Refers to the idea that humans can talk about things that are not physically present or that do not even exist.
Speakers can talk about the past and the future, and can express hopes
and dreams. A human's speech is not limited to here and now.
Displacement is one of the features that separates human language from
other forms of primate communication.
Productivity Refers to the idea that language-users can create and understand novel utterances. Humans are able to produce an
unlimited amount of utterances. Also related to productivity is the
concept of grammatical patterning, which facilitates the use and
comprehension of language. Language is not stagnant, but is constantly
changing. New idioms are created all the time and the meaning of
signals can vary depending on the context and situation.
Traditional transmission Also called cultural transmission. While humans are born with innate language capabilities, language is
learned after birth in a social setting. Children learn how to speak
by interacting with experienced language users. Language and culture
are woven together.
Duality of patterning Meaningful messages are made up of distinct smaller meaningful units (words and morphemes) which
themselves are made up of distinct smaller, meaningless units
Prevarication Prevarication is the ability to lie or deceive. When using language, humans can make false or meaningless statements.
Reflexiveness Humans can use language to talk about language.
Learnability Language is teachable and learnable. In the same way as a speaker learns their first language, the speaker is able to
learn other languages. It is worth noting that young children learn
language with competence and ease; however, language acquisition is
constrained by a critical period such that it becomes more difficult
once children pass a certain age.
One can conceive of circumstances in which traditional transmission and learnability did not fully apply, for instance an artificially created sentient species could have the grammar and basic vocabulary of a language "hard coded" into them from the start. But if that language were a true language it would still allow them to conceive of other possible languages even if physical or mental constraints stopped them being able to use them. And any true language must be able to coin new ways to describe new circumstances, so even a hard coded or genetically transmitted language would include some learned components.
You also asked what series of events could be observed to decide that the aliens possessed true language.
Even if the observer had not yet learned the language, it would be possible to directly observe cultural transmission happening, for example in childrearing. It might also be possible to directly observe prevarication, e.g. a member of the species being lured into an ambush. If the observer saw one member of the species see an unusual situation of potential threat or benefit, then go to tell others who promptly took the actions needed to cope with exactly that situation, then the observer could deduce that their language could deal with displacement. If the observer were allowed to bend the rule of scientific "invisibility", then whether the language possessed productivity might be tested by depositing some completely novel objects near one member of the species and seeing if there was evidence that they had communicated the nature of the object to others, for instance if another member of the species made a picture of that object that could only have come from a verbal description.
Observing duality of patterning and reflexiveness might have to wait until the observer had learned the language - or, more likely, would be part of the process of the observer learning that language.