This is always confusing; the terms are often used interchangeably.
Here’s what I’ve gleaned over the years:
If it’s the intended fill point for a system, it’s a “header”. Remember when almost every car had a cap on the radiator? The cap was on the “header tank”; that’s where you filled the system. Of course lots of radiators don’t have caps, so the radiator’s tanks are just…tanks. Not header tanks.
If you have a V12 Jag you fill the system via a tube at the front of the engine. That’s a header, even though there isn’t a tank. Some 6-cylinder models have a fill point at the thermostat housing and this, too, is a header…but without a tank.
If the system header is a separate vessel entirely, it’s a “header tank”
In other arenas (boats, industrial, farm equipment, etc) you can occasionally see reference to filling a fluid system “at the header”, even if there isn’t a tank involved.
Any tank, apart from the header tank, provided for the purpose of giving hot coolant a place to expand into, can rightfully called an expansion tank. If pressurized with the system the name is probably best left simply as “expansion tank”. If vented to atmosphere it’s probably best described as an “overflow tank” (as that’s where excess coolant overflows to) or “atmospheric tank” (because it isn’t under pressure).
Well, that’s how I see it. I’m not contending that mine are the final words on the matter