But the note you can attach to a zap, that also appears in Nostr clients. Is that a 'note' in the same way a post on nostr is a note? Or is that another note that just somehow appears here?
Discussion
Here is some reading on it:
https://github.com/nostr-protocol/nips/blob/master/57.md
Yes they are kind of nostr note.
I'm pretty sure you can add a message to any lightning network transaction because my wallet of Satoshi would show me what was written in zaps when I was using that to receive zaps