Math in high school is primarily about memorizing and applying set recipes for problems. Math at (a serious) college level has a large proof-theoretic component: prove theorems not solve problems. Math research still involves solving problems, and proving theorems but it has a novel dimension: stating conjectures & theorem, and most importantly the search for the 'right' definitions.
If math in high school is like playing a game according to a set of rules, math in college is like devising optimal strategies within the confines of the rules of the game [actually this is more than an analogy!] than math research involves not just playing the game and finding the optimal strategy but coming up with novel games, with well-chosen rules that are simulataneously 'simple & elegant' yet produce 'interesting, complex, beautiful' behaviour.
I thought not cuz i didn't see why that'd be desideratum. You mean a good definition is so canonical that when you read it you don't even consider other formulations?