I still think you should charge for this service, depending on the conversation you had with the boarder when you texted her and let her know about the injury. If you said something like "That's okay, I'm already here and I can handle it" ....the boarder may have reasonably assumed the service was free. If you said nothing of the kind, and the boarder *asked* you to handle the horse for the vet and take care of it, I think you should charge a reasonable fee.
I would start the clock at when the vet actually showed up, I would not charge for waiting time, even considering it's after hours. If you normally charge a fee for handling for vet or farrier, I would let that fee cover the first hour, and then figure out an hourly rate for your time after that. $50. in total seems to be totally reasonable to me.
If this is a boarder that you have a good relationship with, explain the situation to her and ask her what she thinks is fair.