Apologies for my brevity and haste. Cold-storage is for me where it takes significant time and effort to do a transaction. It IS a kind of wallet to use the broad term but, just as in meatspace we make a distinction between a wallet and a safe or vault, I prefer not to use the word wallet with cold storage.
A wallet I think is more day to day, close at hand, with you so you can make quick purchases and receive small amounts. A hardware wallet can be employed here but probably isn't necessary if we're talking amounts that you'd keep in a meatspace wallet.
Hope it helps and thanks for the question. I prefer to write more to be as clear as possible so feel free to ask more if you still have questions.