My definition of a cold wallet would be one that the keys are created offline and you only sign transactions through an airgap (offline).
Cold wallet is simply a wallet that isn't used regularly. You can air gap it for even more security. I guess some do consider it to be air gaped by definition. I never considered it as such.
Looking up "cold wallet" it does seem the majority consider it "offline" so I seem to be misusing the term. I've always considered it something in other than a hot wallet that may or may not be airgapped/offline.
Once you set this up it is pretty much air gapped as you have no need to login.