I live in Canada, where a pan is shallow and has 1 long handle and a pot is deep and typically has 2 small handles. A pot isn’t a pan, although you can get crossovers like a saucepan which is typically deep like a pot but has a single long handle like a pan. If it’s not shallow it isn’t a pan. Pans can include frying pans, skillets, saute pans, even a wok would be considered a pan. Pans are for cooking at high heat. Pots are for boiling things or for preparing something that’s mostly liquid: soups, stews, sauces, etc. You can also have roasting pans or cake pans for use in the oven, but once again, the key thing is they have shallow sides compared to the bottom.
To me, a pot being a subtype of pan is like saying a knife is just a subtype of spoon. They’re completely different things.