I see these things differently. You and me have already discussed all relevant stuff and I think there is no use to start our dispute again here. You stayed with own opinion but me with mine.
As much as i value your opinion, this has nothing to do with point of view.
It is about terminology.
People refer to the internet as "the web" all the time.
Even if that is
their opinion, it is simply wrong and doesn't make it right.
Same applies to wallets.
If you use the most commonly used taxonomy - storage of private keys - you either have online wallets (e.g. software-, web- or browser-based wallets) or cold-/offline wallets (paper wallets, air-gapped setups) and depending on the PoV "hybrid" ones (e.g. hardware wallets, which is quite controversial where they belong to).
You know even Ledger is not a pure hardware wallet. It has some embedded software (like OS for example) which allows it to operate.
Wait.. are you really trying to argue that hardware needs software to be usable?