[throwaway81523]

No it's pretty standard. Peano's axioms from the 19th century had an induction axiom in second order logic, since it quantified over predicates. Peano arithmetic (PA), also called first order arithmetic, came later. It is a first order theory whose induction axioms are an infinite schema. To confuse things further, second-order arithmetic (SOA) is also a first order theory, whose objects are naturals and sets of naturals.

[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.