I agree with the claim that problem solving is essential to learning mathematics. And I have had the exact same experience where I did well on math exams by ignoring most of the theory and proofs of main results, and focusing solely on examples and problem solving.
However, I think straight-up memorizing definitions and theorem statements is really useful for problem-solving/exam prep, just so they're at your fingertips. There's no way you're passing a real analysis exam if you can't regurgitate the epsilon-delta definition of a limit in your sleep.
What seems to occur (at least for me) is that you naturally memorize all of these things in a somewhat inefficient fashion by doing problems. If the concept gets used in enough problems, it slowly burrows its way into your memory - and this is a very durable kind of memory, as you point out. But I do think for things like graduate school qualifying exams you can "juice" the process by explicitly memorizing core material.
Probably it's not as useful for doing research, though.
I never remember explicitly learning the epsilon-delta definition. By the time I had worked through enough problems, read enough books etc I just knew it.
I saw other students learning definitions etc off by heart and to me it seemed liked they were doing it because they hadn't really understood the material and it would get some them marks. It was probably the right exam strategy if you didn't deeply understand the material to optimise your marks and get a reasonable pass mark but I don't think it was the right way to really learn mathematics or get one of the top marks.
I did learn and test myself on the structure of some of the more complicated proofs in my finals I mostly revised by doing old papers though.
(Feel I should offer some credentials but also don't want to brag, but feel I am about as qualified as one can be to talk about doing extremely well in maths exams :-))
However, I think straight-up memorizing definitions and theorem statements is really useful for problem-solving/exam prep, just so they're at your fingertips. There's no way you're passing a real analysis exam if you can't regurgitate the epsilon-delta definition of a limit in your sleep.
What seems to occur (at least for me) is that you naturally memorize all of these things in a somewhat inefficient fashion by doing problems. If the concept gets used in enough problems, it slowly burrows its way into your memory - and this is a very durable kind of memory, as you point out. But I do think for things like graduate school qualifying exams you can "juice" the process by explicitly memorizing core material.
Probably it's not as useful for doing research, though.