I am buying a new iPhone 12 Mini for my mother who lives in Canada but spends winters in California. I will buying the phone in California. When I check Apple online, I see a different price (lower) for a carrier branded phone, and a higher price for an 'any carrier' phone.

I thought all phones sold by Apple are unlocked regardless of carrier selection - or are the carrier phones locked even if you put in a different carrier SIM card? I would prefer not to waste the extra money on a non-carrier phone, but I want to make sure of maximum compatibility across frequencies. She uses a Fido/Rogers SIM in Canada and usually Verizon in California.

I just want to make sure that if I buy the ATT carrier phone from Apple, that it won't be locked or become locked to a specific carrier due to the difference in price as the phone will never be activated on ATT or TMobile due to the poor service they offer in her area.