It’s easier to remember it as a bit inverter, or as boolean “not equal”, IMO. And “exclusive or” pretty much already spells out what it does.

Regarding the first one I might be biased by learning XOR from https://www.chiark.greenend.org.uk/~sgtatham/quasiblog/xor/#..., however.

Red pill VS Blue pill, that's how I use to memorise it. You must choose one.

Now, it just means 'not equal', 'different', 'not same' etc.

I use this to remember what it does and also like thinking of XOR as an operation you can apply with a mask to flip bits (the linked article mentions this).

Sounds like NAND.

that would just be "not both"