This is a good point. Your grandparent's original distinction was that 1+1=2, just as 2^1=2, follows from a definition while 0^0=1 is immediately defined. Howev

This is a good point. Your grandparent's original distinction was that 1+1=2, just as 2^1=2, follows from a definition while 0^0=1 is immediately defined. However, the parent's still valid point was clearly that even the plus operator is in the larger sense just as arbitrary as the definition of 0^0, concurring with the overall theme of this thread that mathematics ultimately gives no credibility to intuition.