Why didn't you explain to her that different cutting heights aren't going to effect how fast it grows and that if you cut lower you would either have to bag or double cut which would be an additional charge.
Sounds to me like you just didn't communicate and are very defensive with the "this is how its gonna be then" attitude. Most people don't know much about grass, that is why they hire someone. You should have took the time to explain the consequences to the contact better and let them decide.
I'd chalk this one up as a lesson. Communicate better next time and learn why you don't cut grass down to nothing so you can educate them on it, then let them make the decision. If you don't educate them, they are gonna just say do it, cause they are the ones paying you to do it.