First-class polymorphism is what System F gives you: functions from types to values.

Second-class polymorphism is what Damas-Milner gives you: let-bound identifiers may admit more than one type, in which case every type they admit is subsumed by a type schema.

Second-class polymorphism rules out polymorphic recursion if you consider every recursive definition as syntactic sugar for applying a fixed point combinator to some expression of type `a -> a`, for whatever monotype `a`.

That's a new turn of the phrase for me. I've only ever heard the expression "second-class parametric polymorphism" used in reference to enforcing predicativity (which does not rule out polymorphic recursion)

Ah, I might be wrong, then. Pretend I said “let” instead of “second-class”.