Physicist here. Classical error correction may not always be a straight up repetition code, but the concept of redundancy of information still applies (like parity checks).

In a nutshell, in quantum error correction you cannot use redundancy because of the no-cloning theorem, so instead you embed the qubit subspace in a larger space (using more qubits) such that when correctable errors happen the embedded subspace moves to a different "location" in the larger space. When this happens it can be detected and the subspace can be brought back without affecting the states within the subspace, so the quantum information is preserved.

You are correct in the details, but not the distinction. This is exactly how classical error correction works as well.

This happens to be the same way that classical error correction works, but quantum.

Just an example to expand on what others are saying: in the N^2-qubit Shor code, the X information is recorded redundantly in N disjoint sets of N qubits each, and the Z information is recorded redundantly in a different partitioning of N disjoint sets of N qubits each. You could literally have N observers each make separate measurements on disjoint regions of space and all access the X information about the qubit. And likewise for Z. In that sense it's a repetition code.