For me, I use socket if it goes over the fastener and bit if it goes into the fastener.
There are 6-pt hex and 12-pt double-hex sockets, as well as 6-pt E-Torx sockets.
There are 6-pt hex (aka Allen) bits and 12-pt double-hex bits (maybe?), as well as 6-pt Torx bits.
There is 12-pt triple-square (aka XZN) bits as well. The double hex vs triple square is exactly as the geometry draws them out. They are both twelve points, but the flats and angles are not the same.
Notice how if you extend the flat of one pt, it lines up with the flat either 2 pts or 3 pts away. If every other, it's a double hex; if every third, it's a triple square.
I've never seen triple-square sockets. I've also never actually noted a double-hex bit.
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