Hi Badboy,
>However, this information gives me the impression that there wasn’t a "correct" designation as such, just a battle of minds, between Messerschmitt’s natural desire to have his aircraft named after him, and ...
... Milch's desire to have it his way, actually. BFW was more or less controlled by Messerschmitt at the time, but Milch had been Mr. Luft Hansa when some Messerschmitt-built passenger planes crashed, and he was Mr. RLM now and hadn't forgotten the bitter battles fought over the crashes.
Ironically, in 1938 Messerschmitt acquired BFW, founding the Messerschmitt AG, so "Me" became official even with the RLM where Milch was sharing his power with Udet by then.
Regards,
Henning (HoHun)