Wikipedia (https://en.m.wikipedia.org/wiki/Smooth_morphism) defines a smooth morphism of schemes as one that is locally finitely presented (lft) and flat with regular (=smooth) geometric fibers. But the article goes on to say that this is equivalent to the geometric fibers being smooth varieties, which I don't agree with: varieties are separated but nothing in the first definition rules out the fibers being non-separated. Indeed the affine line with two origins is locally the usual affine line, so I want it to be smooth.
This issue comes up in Vakil's notes too: in theorem 25.2.2, the following definitions are claimed to be equivalent:
(ii) a smooth morphism is lft, flat of relative dimension n, and the sheaf of relative differentials is locally free of rank n
(iii/iv) a smooth morphism is lft and flat with (geometric) fibers smooth (in the sense of the Jacobian criterion) varieties of dimension n.
But again the first definition allows affine line with two origins over a field, whereas the second rules it out.
My guess is that everyone really means k-scheme instead of variety, but am I missing something?