How to join seams between hardiebacker and drywall in a corner?
Don't worry as much about the drywall mud.

As you said, fibatape #4. Do that first.
For #3, set your bead in drywall mud. Then, finish the drywall side in drywall mud and the upper side above the showerhead in drywall mud. Feather the lower portion of the bead out with thinset, just to keep everything on plane. Personally, I like to also paper tape the bead to the drywall in moist areas to prevent cracking later. That is not a requirement though, just a personal preference.

Seam #1 &amp;#2 just finish with drywall mud &amp; paper tape.

Looks good man, keep pushing forward.