https://s31.postimg.org/4kozbjqbf/x_y_lines.png
that highlights 0^y=0 in red and x^0=1 in purple. Now it's even more obvious why any single answer is insufficient: 0^0 is a vertical line, not a single point!
Your re-iterating your point makes me wonder if you're actually hearing and understanding the part about choices and definitions trumping logic. If we define 0^0=1, then it doesn't matter what arguments we have, it doesn't matter that there are 3 or more right answers. The right answer can be, and in other parts of math, is what we define it to be, not what makes the most sense.
0^0 is defined as 1 (sometimes) because that has utility and consistency for some other expressions, formulas, operations, etc. It's not because it's right. It's not because it makes sense. It's because we decide to set 0^0=1. It's for convenience.
Now, some sources online say that 0^0 is indeterminate, not equal to one. Other sources online say that 0^0 is 1.
http://mathworld.wolfram.com/Power.html https://en.wikipedia.org/wiki/Indeterminate_form
https://cs.uwaterloo.ca/~alopez-o/math-faq/mathtext/node14.h... https://www.google.com/search?q=0%5E0 https://betterexplained.com/articles/understanding-exponents...
It's a beautiful example, btw, thank you for contributing the visualization!