[galimaufry]

I've heard it called a "non-central example". That has a negative connotation though, I wish there was a more neutral term. The statement "all natural numbers are greater than 1" is false, and it's not helpful to object that 0 and 1 are noncentral examples.

(It's not a motte-and-bailey, that's about a particular sort of shifting goalposts)

[tines]

Nitpick, I don't think 0 is a natural number

[whatshisface]

Zero is sometimes a natural number and sometimes not. N is written as N_+ or N_0 sometimes to indicate which one is being used.

[diffeomorphism]

Matter of taste/notation. Both definitions 0,1,2,... and 1,2,3,... are very, very common.

[leadingthenet]

1 is still not strictly greater than 1, though.