The fun we used to have playing violin (because I bought a violin and am learning to play it along with her) has turned into a nagging session of me trying to convince her that she wants to play/practice. The teacher tells her that sometimes you have to do things you don't want to do, and she must just do these things that she finds hard so she can get through them and move on to more difficult techniques. She has gone from enjoying it, to feeling like she's trapped and it isn't fun.
I think lots of teachers have it in mind that the primary goal of doing anything is to make progress. That means anything that gets in the way of progress needs to be pushed past.
It seems many music teachers tend to see that progress must go along a straight path through a set of skills. You can't go onto skill 4 until skill 3 has been mastered.
But not all musical skills are linked in a line like that. And often playing something a child wants to play that is a lot more advanced can be motivating to learn the skills, rather than learning the skills so one day they can play what they want.
You could tell the teacher that the most important thing is for your daughter to enjoy whatever she's doing. If she hits upon something that she's not ready to push past, then she needs more options of things to do in the meantime until she is ready.
If the teacher can't picture that, I'd find a teacher who was more flexible about how her students learned. Or maybe your daughter needs a break. Though your daughter also needs to know that that particular teacher isn't the only path to playing the violin. She needs to be aware of the options she has beyond quitting and doing what that teacher tells her.
I have a feeling that if I don't put any pressure on her to practice, she will either decide to quit (which I hope she doesn't) or start practicing on her own.
But can't there be a 3rd choice? That she continue the lessons without practice? Perhaps she feels that what she gets from the lessons is enough for what she needs. Is there a specific goal that everyone must head toward as soon as they pick up an instrument? Isn't playing well enough for your own pleasure as important -- well, more important really! -- than playing for other's pleasure? If you pay for lessons until she leaves home and she has good feelings about the instrument but you aren't satisfied with how far she progressed, was the money wasted? Were you paying to create a certain level of performer or were you paying for the opportunity for her to get what she needed and wanted out of the instrument?
Rather than an established ensemble where she needs to be able to play at a particular level, perhaps she'd like playing for fun with a group of kids or adults who get together just to play. That way the purpose is in the enjoyment of making music rather than performing to a certain standard that an audience would enjoy hearing.
She will practice not because the pressure is gone but because she sees something that's important to her that she wants to achieve. I think too often as soon as lessons start the goal becomes performing for an audience when the goal should be enjoying the instrument and what can be done with it. Kids get to see and hear and experience professionals or those who are on the path (self-chosen or socially driven) to professional. They don't get to experience families or informal groups just playing for fun. Which I think is way more important! And a lot more fun too :-) And if a kid is having fun, then they're far more likely to play and mess around on their own than if they feel pressured to work towards something that they don't really want.