If I understand what you want, the short answer is your #1 - a single Player under which is a single Instrument or staff, containing both parts. I’ll explain this as I understand it, and as it appears to work for me. But bear in mind that I’m just another user, not a Steinberg rep.
The first thing i had to learn was the nomenclature. On the left panel, the name in white is the Player (which never appears in the part or the score), the name(s) in blue is/are the instrument(s) that Player will play and also the staff name(s) in the score, and the Layout Name in the panel on the right is the name that will appear on the part (not the score.)
It’s critical to understand that the staff names (full and abbreviated) that you want appear in the SCORE are assigned to the Instrument name (not the Player name) on the left, and the name you want appear on the PART is the Layout Name over on the right.
To get what I think you want, first delete one of the two instruments under the Player name on the left. The instrument, or staff, that remains will be where both flute parts will go. Then hover over that remaining instrument/staff until the arrow appears, click on the arrow, select Edit Name, and enter something like Flutes, or Flutes 1-2, in the Full Name field and something like Flts. in the Abbreviated Name. Again those are the names that will appear in the score. (The Player name, labeled Flute 1 and Flute 2, doesn’t appear in the part or the score, so it doesn’t matter what it’s called.)
Now double-click on the Layout Name on the right, and name that whatever you want to appear on the part (not the score). Again, you’ll probably want that to be Flutes, Flutes 1-2, or something similar.
Now enter both flute parts on the single flute instrument/staff you have set up on the left. That should give you what I think you want.
This all confused the heck out of me when I started, but once I got used to it, it makes perfect sense.
Good luck!