(1.) I agree that he was probably trying to convey "the duration of the light 'pulse' from the flash" when he said "the speed of light.

As the light pulse is of short duration it will, in all cases when the shutter is fully open, have a duration shorter than the minimum possible opening time. This was not true of ye olde magnesium based flash bulbs which had a duration closer to the fully open shutter times of modern roller blind shutters.

Note that the phrases above - "when the shutter is fully open" and "roller blind shutter" are both relevant to this answer. Most SLRs and DSLRs have roller blind shutters which are only ever fully open below some relatively low speed - mayne in the 1/120s - 1/250s range. Firing a flash when they are not fully open will result in a flash illuminated area with dark areas on one or both sides. A focal plane shutter tends to be much faster - in some cases faster than 1/1000s, but these are essentially only ever found on old or specialist cameras.

(2.) Expand statement along the lines - "It was only the amount of light let into the lens from the burst produced by the flash that determined the exposure - not the time that the shutter was opened for as this was far longer than the time taken for the flash burst, but there was no other light received before or after the burst.