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I have to agree that it does get confusing. In my family home we had a Water Boiler that heated water for circulating hot water for heating. The system was changed out to steam heat and then we had Steam Boiler. Both were made by the same company.
The ususal definition is that in a boiler you allow the liquid, normally a single liquid, to vaporize by supplying the latent heat and providing a means to separate the vapor and liquid.
In a heater there is no attempt to vaporize the liquid and is even prevented from vaporizing. There are also exceptions to this as we have single effect evaporators where the steam chest that boils off water is called a heater instead of a reboiler because the bulk of the feed to the evaporator is only getting heated a few degrees.
The term reboiler is usually reserved for when you are using a heating medium, Steam, Hot Oil, etc, to vaporize a liquid or mixture of liquids. Though you can have reboilers that don’t allow the liquids to boil/vaporize in the reboiler, as on some distillation columns or evaporators.
This can get even confusing in one situation we have. We have direct fired boilers, primary, vaporizing a heating medium (Therminol) which in turn is used to vaporize Therminol in other smaller reboilers, secondary, used to regulate the temperature at specific points in a process. On the primary side we only use the latent heat of the vaporized primary Therminol as the condensate is returned to the primary boilers within a degree or so of the boiling point. In this system there is also a direct fired heater that just heats liquid Therminol which is then allowed to vaporize by means of a flash drum. The heater/flash drum system was installed to have a system with a larger turndown ratio than a vaporizing boiler.
It gets even more confusing as you go from industry to industry and country to country.