In long mode, compilers will typically emit `xor eax, eax`, as it only needs 2 bytes: The opcode and modrm byte. `xor ax, ax` takes 3 bytes due to the operand size override prefix (0x66), and `xor rax, rax` takes 3 bytes due to the REX.W prefix. `xor eax, eax` will still clear the full 64-bit register.

Shorter basically means you can fit more in instruction cache, which should in theory improve performance marginally.

Size isn’t everything. You should start by reading the manual for your CPU to see what it advises. The micro-architecture may treat only one of the sequences specially. For modern x64, I think that indeed is the shorter xor sequence, where, internally, the CPU just renames the register to a register that always contains zero, making the instruction independent of any earlier instructions using eax.

IIRC, Intel said a mov was the way to go for some now ancient x86 CPUs, though.