Hmm, I agree that it can be hard to be sure what Header level one is seeing/using (sometimes I quickly try applying a level to my current header to see if it changes as a form of diagnoses/identification 
). I think the issue of navigability for readers is also important, but separate.
You have titled this Feature Request rather broadly, and within it I only really identify the one general request: make it more clear what Headers users are Applying, and which level their current text is at. If that is the extent of the feature request, can I suggest changing the Title to more clearly reflect that? If there is more that needs to be done for “creating a structured document”, let’s elaborate further. 
As to the specific request to make Headers more clearly identified, both for authors and readers, I think your suggestion of putting the e.g. H1, H2, etc. next to the text is good, except I’d say it should only show on-hover/mouseover, just as Collapse and Anchor Link already do. To me this would completely solve the ambiguity issue in a very simple, clean way.
For readers (and, to some degree, editors) to better understand where they are within text as they read, e.g. say you’ve scrolled past the H2 that identifies the section you’re in, so you can’t easily mouseover it to know where you are in hierarchy, I think the best solution is a floating Table of Contents (which I describe in my feature request above). An alternative could be to have some “level” indicator in e.g. the left margin/gutter, outside of the text, which just shows a little number or maybe e.g. “H2”, but it starts to make things look busy I think. It could perhaps only show if you move your cursor outside the text, or maybe if you hold Alt as-in my more broad “Modal block vs. text editing” concept described here (which is to say only the concept of holding a modifier key to invoke additional info/context, so this is partly just an excuse to link my write-up ):