[Tarski's theorem] asserts that there's no first-order description of what statements are true in a system that lies in the system.

You'd need be careful about what "system" or "description" are you talking about, since you can weakly represent, say, the set of statements true in Robinson arithmetic (which is recursively enumerable), in Robinson arithmetic. There is a formula s(-) with one free variable such that

Q |- s(gn(f)) iff Q |- f

where gn(f) is the Gödel number of (arbitrary) statement f and Q is Robinson arithmetic.