I guess you meant "cold" wallet sitting on air-gapped device that was never been online and in the future will never go online.
There is no difference between an offline- and a cold wallet.
"Offline" does not only refer to the internet, but to any communication interface (wifi, bluetooth, ..).
Therefore each cold wallet is indeed an offline wallet and each offline wallet is a cold wallet.
Some users are confused with two concept "off-line" device and "air-gapped" one and think if they turn off their WiFi or pulled Ethernet cable that would be enough.
And some users are confused with the concept of computers believing that it is some sort of magic machine.
I don't see any relevance to that topic.
If they simply just disconnect their internet connection, it doesn't make their online wallet an offline-/cold wallet.
Just because their software is running on hardware, it also doesn't make their software wallet a hardware wallet..