In setup mode, using the Orchestral > Modern template, I’m altering the names of the instruments, starting at the top. After adding Flute as doubler, it’s calling that player Piccolo + Flute 1, and the actual Flute 1 is now Flute 2.
Yes, this is correct: Dorico (re)numbers the instruments if there are more than one of the same kind, also if in another player. What you would desire they where named exactly?
I see. For this I suggest to make the setup as in the left side of the picture , and then use the function Use Custom Player Order in the Layout Options, as on the right of the picture: