That’s a NAND gate (back is flat like the AND, front has the little ball on the output to indicate it’s negated). The back of the OR/NOR is curvy.

AND is “Only both shall pass” hence NAND is “No more than one one shall pass” (which are the other logical options).

As other pointed out what you’re describing is the negation of a XOR (which is “no more and no less than one shall pass”), i.e. NXOR.

OR would be “One or both shall pass” hence NOR would be “None shall pass”.