I am struggling to understand what is different from 'Godel incompleteness' (GI) and call it 'ordinary completeness' (OC). Godel showed FOL to be complete. Yet, Godel's theorems establish that the language Q (an axiomatic extension of FOL) is incomplete.
What distinguishes GI from OC in terms of FOL?
The completeness of first-order logic says that any sentence that is true in all models is provable. This is still true in a system like Robinson's system $Q$ that can model enough of natural number arithmetic for Gödel's incompleteness theorems to apply. The incompleteness theorems imply that there are sentences that are true in the standard model of arithmetic that cannot be proved: these sentences, e.g., a sentence $\mathrm{Con}(Q)$ that asserts the consistency of $Q$, will be false in certain non-standard models. Non-standard models include non-standard "numbers" that are greater than any number that can be obtained from $0$ by adding $1$ a finite number of times. A non-standard model can satisfy the axioms of $Q$, while admitting non-standard numbers that behave as if they were the Gödel number of a proof of the inconsistency of $Q$.