Hob is only a “cooktop”. It always means just the cooking surface, whether part of a free standing cooker (stove) or built into a counter top, it’s a hob.
It goes back to at least as early as the 1600s and referred to a flat shelf above or beside a fireplace that was used to either keep food warm or cook it.
Old pre electric irons were also heated on a special hob.
It became the term used for the surface of a range and then continued to be used for modern cooking appliances using gas and electricity.
The term “burner” is never used in association with electric cookers. You would confuse people by using the term as it doesn’t make sense here. It’s always “ring” and that’s carried through to ceramic hobs and induction to describe zones. If you were ordering a spare part it would be a “cooker element” or “ring”
It’s also the normal phrase used in reference to gas rather than burners, you’ll hear “ring”.
The term burner is very technical here. Like you might actually order a burner as a spare part, but you’d never really refer to cooking on one. They’re often called a ring burner technically too because of their shape.
The term furnace here isn’t used either. It applies only to big industrial devices for melting metal or incinerating waste.
The device that heats a central heating system is a “boiler” (even though it doesn’t boil). So you’ll have a gas boiler, an oil fired boiler, a back boiler (in a fireplace) etc etc
A water heater can be called a water heater, the water heating, an immersion and often instantaneous water heaters are called combi boilers as they are usual a dual purpose unit that burns has to heat either radiators or hot water for taps.
The term faucet isn’t used at all. It’s always tap. Faucet doesn’t really mean anything here - I’m not aware of the term ever being used. It seems to have ended up in US English from old French. The modern french term is robinet. Faucet isn’t used.
