Okay it sounds like you want strangers on the internet to tell you that you are a good babysitter, caregiver and this person the parent of the child, is over reacting and should respect the way you discipline the child.
Here's my question. What difference does that make if we do? Are you going to show this thread to them and say, "See?! You're over reacting! Now you should just believe me!"
How about instead, you, I don't know, talk to THEM? Tell them that you aren't going to watch THEIR child, no matter how much you care about them, if there are such large disagreements about what is and isn't acceptable? Whether we believe you or not or tell you that they are allowed to tell you not to discipline their child that way and you need to just accept that and do things the like or not doesn't really matter. Does it?
Me: 40 pansexual poly.
DH: My husband of 21 yrs and father of 3 teen girls.
DC: LDR of +9 years/former