Declaring 2x2=0+2+2, 2x1=0+2, 2x0=0 while 2xx2=1x2x2, 2xx1=1x2, 2xx0=1 seemed arbitrary.
What helped was learning about negative exponentiation and exponentiation simplification, so 2xx0 = 2xx2 x 2xx(-2) = 2xx2 / 2xx2 = 1.
That said, I clearly still take issue with unintuitive interpretations of "nothing" https://stackoverflow.com/questions/852414/how-to-dynamicall...
Your teacher didn't tell you (or told you, but you didn't recognize it as something valuable and forgot it) that exponentiation to zero is a new definition over exponentiation to positive integers. Exponentiation to positive integers is defined somehow, and that definition says nothing about exponentiation to zero. It is a new definition, not something that you deduce.
The same holds for 0^0 or 0/0 (with some amounts of confusion, lies, and hypocriticism).
Which I found incredibly silly once I leaned about negative exponentiation and could now deduce the pattern.
Similarly, 0/0 became much more tractable to me once I learned L'Hôpital's rule https://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule
2x0 = 2x1 + 2x(-1) = 2-2 = 0
2xx0 = 2xx1 x 2xx(-1) = 2/2 = 1
Or if I were using latex:
2 \times 0 = 1_{+}
However, the 1 of addition is 0 when using the standard notation for integers