The AMU documents I know and an ARMY article by Olympic Medalist Lones W. Wigger, who served there, say “U.S. Army Marksmanship Unit”.
The above is from the late seventies/early eighties time frame. Possibly the name changed over the years.
Readers of this forum know that I think these things do matter. They eventually lead to calling a Belgian cartridge Argentine.
Edited for P.S.: Looking at the current AMU homepage, there is no indication that its name was changed since its creation in 1956 by President Eisenhower.
According to an article by the late Jim Schatz, who also served at AMU, in Small Arms Review 2008, there existed ATMUs (Advanced[!] Marksmanship Training Units) at a number of Army posts. But he also gives no indication that AMU itself ever had a name other than United States Army Marksmanship Unit.