I probably should have said "common" notation; it's certainly what I learned. I think you're right in that many authors prefer ≡ to emphasize that it is an equivalence rather than equality relation.
I don't understand the schism. For example, 11=1 gives the equivalence of two different programs. Whereas 33 mod 4 and 1 are equal in Z_4 ...
It bothered me a lot when I took logics lectures that equality wasn't treated as operator or even just anything within a theory (whereas turnstyle was described, at least).
Yeah, sorry, I didn’t mean to split hairs. They’re definitely both in use. I do plenty of work in Zp and I feel bad every time I don’t write \equiv or the \pmod for that matter :)
Source that both are in use: https://math.stackexchange.com/questions/196081/the-right-wa...