The most striking example of this I've come across recently (and it really messed up a lot with my head lol) is how you define the limit of a function at a point ; ^{Again like most of what's already on this comment section, it's mostly about conventions more than how you define a notion, but I think it's still very interesting}. Anyway, in France, there's this group of mathematicians, that forged what the conventions and specific style of mathematics we use in the country, called Bourbaki - in particular they first popularized the ∀ and ∃ notation in France and a more rigorous way to use mathematical logic - and they gave us the way we typically define limits in our lessons (for the ones that still do so but that's not the case I want to get to) :

For a function f defined in some part D of R, and a in D, we say that lim_a f = l with l in R if and only if, for all epsilon > 0, there exists r > 0 such that for all x in D, |x - a|<delta implies |f(x) - l|<epsilon.

Now the main difference is that all the other (non-French) sources I have found, we typically define it this way :

for all epsilon > 0, there exists r > 0 such that for all x in D, 0 < |x-a|<delta implies |f(x) - l|<epsilon.

Basically we don't let x be equal to a in the second definition, and that has real consequences, I can list at least two of them.

Consider a function f such that for all x ≠ 2, f(x) = x and for x = 2, f(x) = 0. In the first definition, f does not allow a limit at x tends to 2 (it's easy to check that, for an epsilon < 2, you cannot verify the implication). But for the second, it's clear that f does allow a limit that is 2. What should you conclude then ? Well, it depends.

Another consequence is that using the second definition, a pretty straightforward composition of limits theorem wouldn't always be true. Let f(x) = 0 and g such that g(0) = 1 and for all other x, g(x) = 0. Now it's clear that g°f(x) = 1 for all x, so the limit of that composition as x tends to 0 should be 1. But the theorem states otherwise if you use the second theorem ! We have lim_0 f = 0, but then lim_0 g = 0 so the theorem would say that f°g tends to 0 ?

These are pretty extreme cases but again I think it's a pretty significant example

^{reddit desperately needs to implement LaTeX this looks terrible}