If your DD likes her long name, then help her insist on it. If she wants to change it then you'll have to let it go. She might change it back, especially if she's only accepted the shorter version to fit in.
My own name is like Jennifer.
Jenny is a perfectly nice name, it's just not mine. I've politely and firmly pointed that out since childhood (trained by my mother). Some people are affronted at being corrected by a child, but I like to think I helped them learn a lesson about treating all people with respect 
Most people who make a mistake say sorry and I say: 'Don't worry, Jenny is nice, but I prefer Jennifer.'
There are still some people who can't say Jennifer, because they think it sounds unfriendly.
Lots of people who know me say: 'Actually, she prefers Jennifer' on my behalf when I'm not around. That's nice but they don't need to. It doesn't happen that often and it's not meant to be rude.
But there are a few people who insist on Jenny because they think I'm being pretentious
. Anyone who does that gets ignored.