There's a THIRD alien language? I thought there had only been two...
Only two
confirmed alien languages.
There had long been speculation of a third alien language. Fans have attributed this to the fact that both languages were quickly decoded, and the show runners (especially DXC) wanted to provide a third language for the fans to decode. However, since the show ended with "The Devil's Hands Are Idle Playthings", they never got that far, and some suggest that they apparently had other things to consider when they made the films rather than new languages.
The inscription on Leela's bracelet is perhaps the most popular theory of a third alien language of them all. However, too many symbols are missing for it to even have a meaningful translation. It is even hard to suggest what the inscription on her bracelet might even mean.
It is also possible that the third alien language will be far more complicated than the second. As such, it would seem an obvious choice to pick an artificial/constructed language to fit the purpose rather than an "encryption" for Latin characters.
To the topic at hand; it does not make sense for most of the languages seen in "My Three Suns" to be the third alien language, some of them were just gibberish and others were already known established forms of writing. In addition, this is a season 1 episode, probably produced and made before they even knew that the
first alien language would be broken so quickly.
It is thus
very unlikely for this to be a third alien language, this early in production. Their similarities may just be coincidental.
As for the note Leela's parents leave her with: That is in alien language 1. We know that, because it has the exact same symbols, and the translation that her parents provide ("your parents love you very much") matches the note exactly in this language. Your argument that they use it on signs is a moot one, as the background jokes are simply that: background jokes. They are hardly canon or should be taken that seriously.
The note is not a clue to a third alien language.