Let's flip this around and imagine aliens are studying human artifacts. How would they prove there is information in them?
A prelude is to prove that the artifacts were created artificially and deliberately. A lot of modern human artifacts obviously use unnatural materials, colors, shapes so this is not hard to show. You could assume that etching a bunch of lines on a metal plate is a lot of work, so it must surely have some purpose to justify that effort. This is the "backdoor" way to show it. Of course, logically, just because someone deliberately drew some lines doesn't necessarily mean they mean something - it's just that it usually does.
To prove information content, you would generally need to:
- Find patterns that are consistent throughout the known corpus
- Provide a description of the patterns that is significantly shorter than the text itself (so no 10 page grammar to "explain" a 2 page letter)
- Show that the patterns are complex enough that they're unlikely to have come from a natural process
With text intended for human consumption, this is easy to do, because human language has very obvious patterns. It was also selected to be intuitive, so that human babies can easily learn it without instruction. This takes care of 1. Also, human brains are quite limited in computational power, and humans are content to speak/read/write at a limited pace, which makes 2 easy. However, observe that there is a minimal amount of sample text you must collect, otherwise 2 and 3 are impossible and 1 is difficult. For English, grammar can probably be inferred from even a single book, but the bigger problem would be the very diverse vocabulary.
If the aliens tried to examine our network transmissions, hard drives, CDs, DVDs, floppies and so forth, they would probably be very confused however. That stuff is all compressed with efficient (max-entropy) schemes, a lot of it looks flat out like random data as a result, and what isn't random has few recognizable patterns. Imagine the task of zero-knowledge reverse engineering something like 7zip compared to reverse engineering English - a completely different ball game. On top of that, much of the information is also encrypted. In fact, the aliens must solve 4 problems at once:
- Guessing the key
- Guessing the encryption scheme
- Guessing the data encoding
- Guessing the purpose of the encoded data
Each one of these on its own is a challenge, and it's basically made possible by making assumptions about the next level down. Solving all at once seems unimaginable. Imagine you're trying to crack an encrypted file - but you don't even know what encryption scheme to plug your putative passwords in, and even if you guessed those right, you get back a JPEG which you don't know how to interpret. So the "correct" output looks just like the gibberish you get from a wrong password. And even if you did figure out how to decompress JPEG, you still don't know if the contents are intended to be text, or sound, or image, or video or a Sid Meier's Alpha Centauri save game.
At the other end of the spectrum, sometimes we create texts that are specifically intended to be fool proof, and readable without sufficient understanding of even our simple natural language. For example, https://en.wikipedia.org/wiki/Long-term_nuclear_waste_warning_messages describes attempts to create a message about nuclear waste, that would be comprehensible even to people who are not able or willing to go to the trouble of trying to decipher an unknown text. A more mundane example is various signage for multi-lingual audiences (airports) and young children. These should be the easiest ones to figure out, although the problem is that they tend to have low information content (since nobody wants to write complicated things with symbols) so it's hard to gather enough unique corpus to justify any claims about meaning. For example, how would you prove that the appearance of ⛔ is meaningful, and not just an artifact of the manufacturing process?
So, generally speaking, primitive texts will be easy to figure out because presumably they are intended for biological brains that are hard to upgrade or reprogram. For a technologically advanced civilization, there will be more sophisticated encodings intended for machines, and they will be hard to distinguish from gibberish. The more they use computers, the harder it will be to show that their records contain information. To some extent, this might also be the case if they are simply much more intelligent than humans.
To address some specific examples you give:
- The Al-Jazeera logo follows an ancient tradition in Arabic calligraphy. Arabic letters look strange to Westerners even if arranged normally, but if you get past that, the logo is basically a monogram just like the logo used by JRR Tolkien, for example. It is not intended to be "read" but "recognized", and it is rare to write original information in this form (it would be very hard to read). It is more common to use it for texts that are well known, such as verses from the Quran or the names and titles of a Sultan. Therefore the corpus for such calligraphy will be limited - but if it was used more commonly, then analyzing the topology of letters would reveal patterns.
- The made up font is so trivial that you can even tell at a glance it's supposed to be English text. It has overcomplicated letters that are hard to distinguish, but any real analyst would start by labeling each symbol with a number or letter and go from there. It's harder if you have organic variations of the same symbol (handwriting), but the different forms will still be more similar to each other than to other letters. Even if you have alternate forms like Greek
ς, they should still be fairly straightforward - otherwise the alphabet itself would be unusable.
- For your triangular sigil, it is not possible to prove that it is meaningful because your corpus is too small. You would need many more sigils like that, so you could look for patterns like the angle of the lines, their curvature, when they cross each other or not. But with just one sigil, I could invent any number of just-so "meanings" for it, all equally correct and plausible. I could claim that the "real information" is that the sigil is black and white as opposed to other colors, and the lines are just decorative. You can't do anything without more samples.