let see, I don't know these terms very well, because I never been introduced to the right way. But my guess?

Exhaust manifold is a PART of the header right?

Header is the Whole Thing, leading from the Engine's Exhaust outlet to the Front Pipe just before the Catalytic Converter. and that Upper Part of the header, having 4 pipes connected to one end is called the Exhaust manifold.

And do you see your car's header if you open the Cover? Yes because that part is the Manifold OF the HEADER, you can't see the WHOLE HEADER, but at least part of it.

correct me if I'm wrong any1?????

THe Heat/Physics and Cover part. Blackmagic explain the terms right. But I don't think it applied here MUCH. The cover is just simply Concentrate the heat in one place keeping the heat from distribute to the whole engine bay, making it hotter than it is for other components: AC, Intake, and every other part.