"Let alpha and beta be any ordinal numbers, then ordinal exponentiation is defined so that if beta=0 then alpha^beta=1."
Again, no exception for the case when alpha=0.
Now, when you get to analysis, Lim(x,y->0) x^y depends very strongly on *how* x and y go to zero. There's no single, universally-applicable answer there. But, for number theory, combinatorics, and pretty much anything dealing with the naturals or the integers, it's not that complicated.
> Yes, in some subjects, conventions are adopted in those subjects. The practitioners know the convention. Good for them. That doesn't in any way change the fact that 0^0 is undefined.
Would you be willing to discuss the distinction between "conventions" and "definitions"?