That's my (uneducated) understanding, but the speculation is that he didn't pay it himself because then he couldn't deny (to friends/family/fans/donors/voters) that he knew her.
Correct. My understanding is that if Trump had paid Stormy Daniels directly and classified it as a campaign expense, it would have been legal and there would be no case.
As for intent, expect it will turn on the exact language in the charged campaign finance laws. If Trump can find a reason he wasn't required to disclose, there is no felony concealment.
On the whole, a good thing for campaign finance law to have a high profile fight between two well-resourced legal teams. Hopefully it might drive some future reform and better law, as the boundaries are clarified.
That is still a campaign expense and needs to be reported to the state. Reporting on who donated money is a totally separate report.
but if Trump paid directly himself, it wouldn't violate campaign finance laws? Because there is no campaign finance limit on Trump himself.
That said, they still have to prove "Ergo, it's a campaign expense" - any individual is motivated to protect their reputation aside from campaigns.