> What I really want when learning a new area of math is the full motivation for the tricky definition

yes so then you want proofs that actually exercise real machinery instead of playing the shell game of "an X is a Y and a Y is a Z, and has ABC properties, there for X has ABC properties"; you want a proof that goes through the process of using properties ABC to build Y from Z and X from Y (or something akin to that).

definitions aren't for people learning math, they're for people using math ie practising professional mathematicians that are proving more theorems; Hausdorff didn't invent "Hausdorff spaces", he used/worked with various properties of topological spaces and then when the next person came along and needed to right another paper on top, that person invented "Hausdorff space".