This can be embarrassing, but most people are very understanding. When I was pregnant, my childbirth instructor recommended carrying around a jar of pickles. It might be a good idea to carry a small towel for this eventuality, although it is an unlikely scenario. In actuality, few labors start with a huge gush of fluid. Especially if this is your first labor, the baby's head is often engaged, or close to engaged, at the start of labor.
Most labors start with some lower back pain or menstrual-like pelvic cramping.
The contractions are felt as tightening of the uterus but are very mild and are initially irregular. First time mothers have a lot of time to get home and have a meal, rest, and start timing the contractions. The first contraction, or often the first 100 contractions, rarely elicits an exclamation from the mother. They start out very gradually, becoming more frequent and more intense as the hours go by. Indeed, if the first few contractions caused that much discomfort, I would encourage a mother to go to the hospital as this would be unusual.