The video where the cat begs (through the buttons) to go outside in the rain, takes one step outside, comes back in, and immediately complains (through the buttons) about the rain, is pretty convincing to me.
Also interesting is its use of the "angry" button in response to not getting its way, or having something irritating it, or when it's clearly frustrated by not having its needs understood.
Maybe there's more in videos I didn't watch, but I watched 5-6 videos and the cat seems to be pressing buttons randomly. It doesn't seem to correspond to anything it does.
AFIACT, the videos are a kind of progress log for demonstrating the cat's learning in different concepts. The older videos focused on [demonstration of] learning of more "core" concepts ("food", "water", "pets", "outside", etc.); but since these concepts are now fully "taught", the owner seems to feel no more need to film the cat communicating these simpler core messages, except when they form an interesting sequence or larger idea.
Instead, the newer videos focus on learning of more fringe concepts — ideas that "come up" less often — and so progress is slower in these.
(Note that, watching the videos chronologically, the number of buttons the cat has available grows quite a bit over time. I think the owner thinks that the cat now understands the general concept of associating things-to-be-communicated with presses of word-buttons, and so feels that they can now feed them new concepts faster.)
I don't think it's completely random, not some of the time, anyway. That's not to say that it's deliberate, though, either. There's a number of videos in which she'll barely touch or just hover her paw over a button, pause for a while, take it away, and then proceed to push a different button.
Also interesting is its use of the "angry" button in response to not getting its way, or having something irritating it, or when it's clearly frustrated by not having its needs understood.