A lot of people already gave the correct explanation about this mate. You don't connect your cold storage to the internet, you only use them to sign a transaction offline and take the signed transaction to a watch only wallet which it will be broadcasted to the network.
You missed most of the conversation. A hardware wallet that connects to an internet connected device directly is itself connected to the internet. Second an offline wallet is one that is always offline, otherwise the word offline wallet is meaningless.. I don't think you understood me. A mobile device is usually one that is connected to the internet, that is how most people use them daily. Just because you put this mobile device offline today that does not make it an offline wallet!! That was my point.
It can't be offline occasionally, sometimes, or something it must be offline always to be considered that. There is nothing to dispute here.No it is not. For a wallet to be an offline wallet it must be offline all the time. It can't be offline some of the time. If it could, then where do you draw the line? 10% offline time is enough? 20%? 50%? Either it is an offline or online wallet it is a binary choice.
If their definition is correct then all wallets mobile, desktop, extension wallets are also offline wallets when you are not on the internet and online wallets when you are.

It can not be any simpler. An offline wallet is one that is always offline and never connected to the internet. Most mobile wallets don't even let you export a transaction so they can never be offline wallets even if the phone was never connected to the internet. How would they be used?