Not GP but, processing payments absolutely does not require a network connection. Doing so is absolutely not nearly as convenient, but in my adult lifetime it was pretty normal for retailers to pick up a phone, give a customer service rep and/or automated call handler CC info and dollar amounts and get appropriate confirmations.
As for a business without an OS interface not being a "store," that's ridiculous on its face. If that were true, we'd have to call 7/11 or any similar place (like those at most gas stations) convenience "locations with items for sale but not a store, because stores are only places with catalogs in my OS," and "places which sell stuff but aren't stores because rimunroe says they can't be a 'store' without a catalog in their OS."
Touch grass, friend.
I can understand how someone under, say, 30, might not know how commerce happened before the Internet. My 13 year old can't believe there was even once a world without the Internet.