I don't necessarily think all of what you did was right or all was wrong. I think a great way to have handled this would be 1. Ask the director to, in the future, schedule your front-desk time around your privates time, or vice-versa, 2. If an unforeseen situation arises, take the athlete to the mom quickly and ask them both if it would be alright to take a 5-minute break so that you can briefly introduce the new parents to the gym, or 2a. Ask the new parents if they wouldn't mind giving you 10 minutes (or whatever) to finish your private with Susie, and 3. Not give your services for free.
Susie and her mom want you, not Coach X. Susie's mom pays for the work that's done and you deserve to be paid for the service you perform. Susie's mom might have had a tight schedule that day and even when you kept her after to make up for the break, it might not have been the best thing for her situation.
I think her remaining upset after this situation might be something you can call her about and say "Hey, I know you were upset after our last private, is there something else we can do to create a positive situation for Susie?" I wouldn't doubt for a second that Susie probably heard lots of negativity about you on the ride home. Nip all that in the bud.