[cubefox]

It's pretty standard also to talk about "first-order Peano arithmetic" and "second-order Peano arithmetic". This is much more clear but inconsistent with the other usage which you describe.

Moreover, non-logicians don't talk about "first-order" or "second-order" logic at all. They just express the induction axiom in plain English, and in this case it is (as Stewart Shapiro argued) equivalent to the second-order axiom.

[throwaway81523]

Yes. I don't remember who called second-order logic "set theory in sheep's clothing". ;)

[cubefox]

Quine. His justification was quite bad though.