Should you keep a notebook of example problems while going through science and engineering courses?
I find myself constantly being told I should not intentionally jog down example problems but to understand the core logic of a topic.


You're offering a false dichotomy.  If you can't generate concrete working examples, you don't really understand the definitions; conversely, if you can't prove things from the definitions, you don't really understand the examples.  You should both work through examples and understand the definitions; the two approaches strongly reinforce each other.  In particular:


If you are given only examples, you should figure out appropriate formal definitions and core logic yourself.  Aim for both simplicity and generality.  Look for weird corner cases.  Work out new examples that stress-test your definitions.
If you are given only formal definitions and few or no examples, you should work out several new examples yourself.  Aim for examples that both illustrate and stress-test the definitions.  Together your examples should exercise every case, every word, and every symbol in the definitions.  Failure to cover the definitions completely may mean that the definitions can be simplified; success may mean that the definitions can be generalized.


Write everything down.  Fail, revise, repeat.