D.B.
I'm a stepmother, and I understand completely how much you love her and how much you want to influence her for the better. I also understand working with a jealous, even vengeful and perhaps manipulative bio mother. Here's my advice.
Stop calling this child your daughter. You can love her endlessly and jump in front of a train for her. But please, for the sake of everyone, stop referring to her as your daughter. Your husband must parent her in your house, period. If he can work with his ex in any way/shape/form, he should try. If they go to parenting class or mediation or a counselor, even better.
Your stepdaughter should not be calling one parent all the time when she is with the other. My husband had a set time to talk to his daughters, and by and large, that scheduled was followed. If your stepdaughter has her own phone, that should get put away when she visits you. Your husband should make it clear that he's not getting in between daughter and mom, but trying to establish a routine. He should offer a sensible plan for calling his daughter when she's at Mom's. The child needs to be told by both of them that she's not going to play one against the other. Your husband should be the one saying "It's how things are" or "different people have different rules. That's how it is with teachers and employers and college professors - you do what's expected and you learn to adjust." Not you. Your husband.
The most you should say is that the reason there is time to do nails is that the chores are done. Many hands make light work. If everyone works, the job gets done faster, and there's more spare time. If she doesn't do her chores at Dad's, then stepmom isn't doing nails or going for a bike ride. But your answer needs to be, "Talk to your dad." "Your dad will talk to your mom." And especially, "Don't say bad things about your mom in front of me. I won't listen to that. Your mom works hard on your behalf and you will speak about her with respect.
None of this "You don't have to go to your dad's" if that's what you're saying. Your husband and a counselor - or a lawyer - can work out parental alienation. Visitation time is visitation time, period. Changes in schedule get worked out between Mom and Dad without the child being in the middle.
Part of the reason the mom is doing this is that she is jealous of you and feels that it's all Party Time at your house and that you are trying to replace her or even equal her. You have to work harder to prevent that.
This will blow up in your face when she's a tween and a teenager. Get ahold of it now. And yes, walk away from the drama and don't get sucked into it. You're being played here a bit.