What are the names of all possible logic gates with 1-or-2-bit input and 1-bit output?
Most of the two-input gates have a single name (AND, OR, etc.), but single-input gates are named not only after the logical function (NOT), but also after the effect they have on the signal or the function they have in the circuit ("inverter", "((non-)inverting) buffer/driver").

"XNOR" is just easier to pronounce that "NXOR".

The gates marked in red are not widely used, so there are no common names for them.
Implementations actually exist in configurable multi-function gates (74xxx1G57/58/97/98/99), but only as a side effect of the configurability.
The SN74LVC1G97 datasheet describes them as "(N)OR/(N)AND gate with one inverted input", and that is probably the easiest way to understand them: