An interesting update to my post:
Working on the Oboe 2/EH staff I have encountered the scenario in which the first note in the music is played by the doubler, the EH.
So I have edited the Full name of the EH instrument in Edit Names so that the staff label in Page mode reads “Oboe2/English Horn”. No other way to achieve that.
But of course… since the first note is played by the EH, Dorico sees no reason to specify “to EH” or “EH” inside the staff, right before the note. It would no be a problem if the first note was played by Oboe 2, because it is understood that Oboe 2 is the main instrument. And labeling the staff a reverse “English Horn/Oboe 2”, so as to justify that the first note appearing is played by EH with no other warning, would be strange.
So I’ll have to manually add a text saying “to EH” right on the first bar, if I want to avoid that the player attempts playing that note with the oboe.
All this would be easily solved if the Player name, not the instrument name, translated directly as the label of the staff. I think.