After 6 years I still don't understand this

If you approach this from 0-, you get 0/0 (indeterminate). Only if you approach it from 0+ do you get 0. Therefore, because there is a discontinuity here, it would make more sense to define 0^0 using the first limit you provided,

since it is continuous at all points except for that one hole at 0. In other words, both x^-0 is 1 and x^+0 is 1, whereas only 0^+x is 0. For a limit to be defined at a single point, both sides of the limit must be satisfied.

Otherwise, the step function f(x) = floor(x) would equal both 1 and 0 at x=1, when it can only actually exist at one of the two points or the function would not be a function.

Furthermore:

k^x = y
x*ln(k) = ln(y)
ln(k) = ln(y)/x
k = e^(ln(y)/x)
k = y^(1/x)

If x = 0, there is just no possible selection of y that can make k = 0. poor wording