What is a logic gate?
You can define the term "logic gate" to be whatever you want, and no one will force you to change your mind. Each person is free to define the term as best suits their needs.

As a CMOS VLSI designer I tend to think of NAND, NOR, inverters, and transmission gates as being the set of "gates". To me, an XOR is usually a multi-gate circuit. When a manufacturer talks about the number of "gates" in some product they usually mean the number of equivalent 2-input NAND gates.

To George Boole, the AND, OR, and NOT operators were the most primitive logical operators so it would make sense that a person could define the AND, OR, and NOT gates to be the only true "gates".