In the US we usually "rev" a motor that's idling or in neutral, while reserving the term "gun" for rapidly accelerating the engine of a vehicle in gear to significantly increase speed, as you describe in your example. Ps: your driver reminds me of my dad!
Edit: I looked for supporting references but couldn't find any that recognize the distinctions I indicated. These different usages are actual in vernacular AmE though probably not recognized by some, and certainly not absolute. One could use these terms interchangeably and be clearly understood. I wouldn't personally, but hey, I hang out here, with a more discriminating crowd!
Ps: oh yeah, in America we not only love to hold a gun, and to fire guns, we also like to use the word gun, as often as possible, hence, we gun our engines.
Here is a reference from a car fanatics forum illustrating my contention that a subtle distinction does exist between revving an engine and gunning a motor vehicle:
TOMCAT104
Whereas turbo's engine low-end torque might be more useful in daily driving situations (and since BMW seems to have reduce the turbo's lag to insignificance), it might have an advantage over the n.a in that respect. True, turbos cannot replace the joy of reving your engine high and gunning it, but in practice for most daily driving situations, would a turbo be actually a more suitable engine?
from BIMMERFORUMS
GUN verb (used with object), gunned, gunning.
10. to cause (an engine, vehicle, aircraft, etc.) to increase in speed very quickly by increasing the supply of fuel.
Dictionary.com
rev (v.) 1916, from earlier noun (1901), shortening of revolution, in reference to the internal combustion engine. Related: Revved; revving.
gun (v.) " to shoot with a gun," 1620s, from gun (n.); the sense of "to accelerate an engine" is from 1930, from earlier phrase to give (something) the gun. Related: Gunned; gunning.
The Online Etymology Dictionary