What defines the curse type?
Curse type is defined by nothing more and nothing less than having the word “Curse” on the card in the type line.

This is true of other types as well, such as “Attack”. Attack does not mean that the card hurts your opponent, it only means that it has the word “Attack” on the bottom. Mechanically and thematically though, it is better to only have Attack type on cards that hurt your opponent. 

Certain types in Dominion, such as “Action”, do have special rules that come with them. But many other types have no rules. Curse has no special rules, the type doesn’t mean anything at all.

That being said, making another card with the type “Curse” is not good for a fan expansion. This is because “Curse” is both the name of a card and a type of card, and the only card with that type is the card named “Curse”. If another card with that type existed, then cards that said “gain a Curse” would be confusing and ambiguous... does “Gain a Curse” mean gain a card named Curse, or does it mean gain a card with the type Curse? There is no correct answer. That ambiguity is only ok because there is no other card with that type.

So, if you want a different card that will hurt players, whether by having negative VP or by clogging their deck, just leave off that type. You can see how Dark Ages did that thing by introducing Ruins.