YANBU
I have a name I don't like, never have, never will.
And when introducing myself to people I get
a) a hideous version nickname which I absolutely detest.
b) People are LAZY and change the last letter so it is two sylables instead of three. Or they just don't have the courtesy to LISTEN and call me by the correct name.
Pet hate of mine!!!!!
If you call your son x then he should be x. It is courteous to call someone, whether adult OR child by the name they are called. Yes he can accept different versions as he gets older, but right now it is the name YOU have chosen for him.
Just say you would rather he were called 'x'.
And I don't think childminder is being horrible and it is a endearment rather than anything else, but she should still have courtesy to respect parental choice of name, not change it to something she prefers for whatever reason, be that endearment or easier to say.