Defining 0^0=1 has issues. Example: a(n)=1/exp(n), b(n)=1/n. Then both a and b converge to 0, but a^b is always exp(-1) (and hence does not converge to 1).
What you have shown is that the _limiting form_ 0^0 is not always equal to 1. The _value_ 0^0 is still equal to 1.
Similarly, floor(−1/n) converges to −1, which tells us that the _limiting form_ floor(0) is not always equal to 0; but the _value_ floor(0) is still equal to 0. Nobody uses this to argue that the value of floor(0) should be undefined or context-dependent.
But you are right, what I meant is there is no way to define 0^0 maintaining continuity of the power function. Why is this important? Because power is a continuous function otherwise.
Similarly, there is no way to define floor(n) for integers n maintaining continuity of the floor function, even though floor is a continuous function otherwise. We still define floor(n) = n because the meaning of the floor function is more important than its continuity. And so it is with exponentiation at (0, 0).
