There are really only two people that are really in a position to know which movement was the first watch worn on the moon for certain: Neil Armstrong and Edwin (Buzz) Aldrin
It has been documented that Armstrong did not wear his Speedmaster on his walk on the moon. An instrument had malfunctioned during the LEM's Decent and his Speedmaster was left in the LEM to serve as a replacement.
[Armstrong, from the 1969 Technical Debrief - "Now, a preliminary comment (on the EVA Preps) has to do with the longer time that it took than during our simulations. It is attributable to the fact that, when you do simulations of EVA Prep, you have a clean cockpit and you have all the things that you're going to use there in the cockpit and nothing else. In reality, you have a lot of checklists, data, food packages, stowage places filled with odds and ends, binoculars, stop watches, and assorted things, each of which you feel obliged to evaluate as to whether its stowage position is satisfactory for EVA, and whether you might want to change anything from the pre-flight plans. For example, our mission timer was out, and we decided we had better leave one wristwatch inside in case it (the one taken outside) got damaged. We would have at least one working watch to back up the mission timer or to use in place of the mission timer, in case we could not get it going again."] ... [The astronauts each had an Omega Speedmaster Professional.]
Later on in the log the following conversation transpires:
108:54:54 McCandless: Neil, this is Houston. Will you give us hack when you start your chronometer. Over.
108:55:03 Armstrong: Roger.
108:55:08 Aldrin: Give it to them later.
108:55:12 Armstrong: Okay. Okay, let's go to dump.
108:55:17 Aldrin: Dump.
108:55:18 Armstrong: Go to dump. (Long Pause)
108:55:40 Aldrin: Houston, I'll set my watch at 56. Over.
108:55:50 McCandless: Roger.
[Buzz is wearing his watch on his suit sleeve and, apparently, is setting his at 56 minutes after the hour, corresponding to the upcoming Ground Elapsed Time of 108:56.]
108:56:00 Aldrin: 3, 2, 1.
108:56:02 Aldrin: Mark.
[Buzz reopened the dump valve on his mark. This is the start of the EVA.]
[Aldrin - "I'm sure that Neil didn't wear his watch out on the surface. I'm sure he put it with the Velcro strap up in the AOT."]
[Armstrong - "Someone, perhaps in correspondence, asked me about that. And I could not remember, although it seems quite logical, given the mission timer situation, that we would have left one watch inside."]
Thus it is apparent that Aldrin's Speedmaster was the first worn on the moon