As your informant told you, the basic expression here is on your mark (or marks) and derives from foot-racing. It is a the first of (typically) three commands given to runners at the start of the race and orders them to take a position at the starting line—the ‘scratch’ or ‘mark’. Here is an early description, from the 1890 Harvard Advocate:
From my window I often watch the hundred yard sprinters racing down the stretch on Holmes Field. As Mr. Lathrop gives the word “Get on your marks,” they stop their talking and laughing, range themselves along the scratch in the shade beneath the willows and with their long spikes begin pawing holes in the cinder-track to get a purchase for the start. “Set,” says Mr. Lathrop. The squad break in the knees and settle into their braces, with an arm outstretched before and behind. [...] I see a red flash from the pistol. An instant the starters tremble violently and, about as I hear the quick report, the men bound simultaneously forth.
On my mark appears to have evolved from this use in US missile services in the late 1950s. A 1983 historical novel suggests it was in use about 1960 to announce a countdown towards a time-synchronization:
“ATTENTION ALL PERSONNEL.” The launch control officer’s voice broke over the loudspeaker, its flat, Victrola tones carrying across the small valley to their observation post. “TASK EIGHT, COUNTDOWN EVALUATION IS COMPLETE. ON MY MARK WE WILL PICK UP THE COUNT AT T MINUS TWELVE MINUTE. THIS IS PHASE ONE OF THE TERMINAL COUNTDOWN. PHASE ONE WILL BEGIN ON MY MARK. FIVE, FOUR, THREE, TWO, ONE, MARK. T MINUS ELEVEN MINUTES AND FIFTY-FIVE SECONDS.”
This is exactly paralleled by the earliest actual use I have found, from the transcript in NASA’s Results of the Third U.S. Manned Orbital Space Flight October 3, 1962:
01 Z0 0l CC ... on my mark the ground elapsed time will be 1 hour, 30 minutes,
and 10 seconds. Stand by. MARK.
01 30 13 P I am exactly I second slow. Correction, I am 1 second fast
03 11 39 CC Let me give you a G.m.t. time hack; see how we are there at this time.
03 11 42 P Yeah. That's probably all fouled up. Okay. You give it to me.
03 11 46 CC On my mark, I'm 15 27 00— MARK.
It is clear that the racing sequence has been adapted. There is a semantic difference: mark now represents the target rather than the starting point, so it is the speaker’s mark rather than the hearer’s; but essentially the same sequence of commands is issued. ‘On my mark’ signals the hearer that countdown is about to begin; a ‘target’ timevalue (with an optional countdown) replaces the ‘set’ command; and ‘MARK’ replaces ‘Go!’ or a pistol shot. And the on my count example exhibits the same pattern: initial phase marked with On..., readiness phase, and target command.
