What Is a "Reverse" In Bridge?
I agree with your understanding of the situation.  By forcing you to go to the 3-level if you decide you prefer his first suit (diamonds)... well, that's a textbook reverse.  I would suggest that "about 17" points is a fair reading of such a bid.

Since your partner only had 13 points, and your bid should clearly have allowed him to infer that between you you only barely have the balance of the points, it's difficult to see what he hoped to achieve by going to 2S.  You are extremely unlikely to have 5 or more spades after your NT bid, so at best you're finding a marginal 4-4 spade fit, at worst no fit at all.  And that's if you stop bidding there, which of course you are unlikely to, because your partner has just falsely suggested that he has some basis on which to keep the bidding going... 4S sounds about right for an 8 card fit and roughly 24-25 points between you, yes.

I've said it before and I'll probably say it again - send your partner back to basic Bridge school!