I am new to using indicator functions (although I am quite familiar with undergrad-level probability and what an indicator function is).
I am trying to relearn probability using indicator functions and using them where I think they would be appropriate. But I don't want to use them where they're unconventional. I've been searching for examples for the PMF of a discrete distribution for example. Some say $$f_{X}(x) = \dfrac{1}{N}\text{, }x = 1, 2, \dots, N$$ but if I were to write it with indicators, $$f_{X}(x) = \dfrac{1}{N}\mathbf{1}_{x \in \mathbb{Z}^{+}_{\leq N}}(x)\text{, }x\in \mathbb{R}$$ or some strange notation like that. I'm not an expert enough to know which one is more conventional.
Are there probability textbooks that rely on indicator functions (preferably at an undergraduate level or at a M.S. level) when explaining PMFs and PDFs, rather than just tossing them off to the side?
The first form that you have written is way much better!
Exactly as you say, we should "avoid strange notation like that". Most of the time we only use them to shorten some specific localized notation that otherwise would be awkward to write in some other manner, say for example inside a specific integral. Personally I have always avoided them, on purpose.