Let's not mince words. The term pucker factor refers specifically to the instinctive tightening of the anal sphincter during periods of high stress. This automatic attempt at bowel control is quantified along a scale of 1 to 10, with 1 being "loose as a goose" and 10 being the proverbial "You couldn't pull a pencil out of his asshole with a pair of linesman's pliers." There does exist a place beyond Pucker Factor 10 - that being "shitting your britches." In combat, many men skip the pucker factor, immediately drop load, but then report feeling better, often going on to fight vigorously. I've read several veteran's accounts of this response, it seems to be somewhat universal. That's something you didn't see in Saving Private Ryan.
The phrase is said to have its origins as Pilots' Jargon, but is widely disseminated among the American armed forces.