It has always been called "Union Station" right from the day it opened, that's even what the sign says, but it's official name was Los Angeles Union Passenger Terminal, LAUPT. I am not sure exactly when they changed it to LAUS, but I do know the reason. After some of the San Diegans/Pacific Surfliners began running through to Santa Barbara (and eventually SLO), it was no longer a "terminal" serving as the origin and final destination of all trains, it was now just a station (well, technically a depot). Someone noticed the terminology and changed it.
Up until the San Diegans began running through only one train ever ran through Union Station, the Sunset Limited from the station's opening on May 4,1939 until early 1942. The Sunset Limited originated in San Francisco, it was cut back to Los Angeles as a wartime measure. The SF-LA portion of the Sunset got deemed a redundant service and the equipment used for "essential services".