A small nomenclature clarification seems to be in order:

"Headless" means without output. It is often used to describe servers without any attached monitors (the monitor would be the head that it is missing), but can similarly be used to describe browsers that do not even create a window on your desktop. It simply runs as a service that can be scripted.

What you are looking for, I would rather call "chromeless". The browser is there, but it has no visible UI-components outside of the webpage.