The quote that Cocoabiel uses, like she states, is a quote from Seth Dickinson:
This [Aiat] is it, and its purpose is expression, its meaning is the invocation of what it is, its mission is to draw in the mind and make an incision of curiosity and to be that incision which is a question and its own solution, to make one hunger for an answer whose answer is its own wanting. Aiat. This is it, it is that utterance.
“It’s just me.” is the newest Truth to Power entry, so there’s a chance it was Cocoabiel needed a definition before its release and turned to Dickinson’s words. There’s also the fact that TTP (while interesting) rather unreliable narratively, making it a less desirable source in some eyes.
The two are worded differently because, textually, they are from two very different sources. Dickinson’s quote is a word-of-god explanation; he’s answering the question “What does Aiat mean?” As the author of the Books of Sorrow, it’s likely he invented the term as it’s used in the Books, meaning he would have the ‘definite’ definition. Meanwhile, ‘Eris’ (a debated subject) in “It’s just me.” is speaking like Eris typically does:
Do you know what the Hive say when they want to express the inevitability of a thing? When they want to say, it is this way because it could be no other way?
She’s typically more vague and circular when she speaks, and this is an example of that same manner of speaking. Eris doesn’t have the same explanation Dickinson does because she’s a character operating from within the game, while Seth is word-of-god’ing his own definition to us.