That is an age in girls when they can start to get a little more challenging (talking from step mumming experience and observing primary school children) as they start testing boundaries, exerting more independence, some start dealing with hormones.
I have a step daughter and I recognise the challenge of her having different household vibes. At ours, there were more people and younger siblings, there needed to be more structure and more team work. At her mum's she was (and is) treated as a best friend. And that's where it was really tricky. She pushed back so much more with me than my boys ever did. And it's not a super strict household!
Similar would happen on family holidays, when she's away with just her mum they could do whatever they wanted and whenever they wanted, and her mum has always leant into the being best friends thing with her. On our family holidays, when they were younger we had babies and toddlers to factor in and when they got older we'd plan trips and activities for that we'd all enjoy but that means sometimes it's something a younger boy or a dad wants to do that doesn't appeal so much to her (steam train related often!) and she'd be more of a grump about compromising than anyone else ever was.
We're absolutely fine with each other, I'm not an evil step mum, but there were some years (especially when she was 10-12) which were pretty tricky. Actually, even when she was younger and I'd be used to wrangling a toddler, young child and a couple of dogs, it would be the mornings she was with us before school which would be the most challenging. I did things like dropping one son off at breakfast club those mornings and walking the dogs with her and the baby so we got some fresh air and chat before driving to school - that did help her to be better regulated going into school that getting straight into the car in a stress.
But I'm going into the details too much here. I just mostly want to say: I get it, it's tricky, I feel for you. It got better for us but she's also distanced herself as she's got older - Covid and then a very busy social life mainly are the reasons. We've had no falling out, she's invited and welcome at every meal out, holiday, trip, wedding.
My advice? It's tricky. I sort of just stuck it out? I talked with the mum and stepdaughter about why our household needs to work as a team, there are too many people for us all to do what we want when we want, that when I ask the kids to help with something or do something independently e.g. "time to pop your shoes on", I'm not trying to be mean, I just want us to leave the house with shoes on. We had conversations about trust and respect with the other parent early doors.
I think it helped to do some of those school drop off and pick ups as I got to know some of her school mates and their parents and I could talk with her teacher.
I know in my heart of hearts that I am a more peripherary character in her life than lots of other people. I'm like an aunt and that's OK. I also know that I've been there for her throughout, I've been steady and constant, I've been reliable and helpful, I've ferried her around her hobbies and activities, I've watched all her concerts and sports matches, I've made sure her childhood birthdays with us were special even though all her friend parties were with her mum - we still had cake and candles, balloons and a family treat at ours, she's included in my mum's answer to how many grandchildren she has. Where am I going with this? Why do I think this is helpful for you to hear? Um, I guess because I want to say that you're not some big bad evil stepmum for having a different parenting style to her mum.
It's sad that it sounds like your husband isn't really supporting you.
Oh I know what I wanted to say too - one of my best mates had a similar but different upbringing to my stepdaughter - she lived mostly with her mum and would see her dad and stepmum every other weekend. Her mum's fun and also wants that best friend with her only daughter kind of relationship. Her stepmum's more down to earth. She tells me that she gets different things that she needs from adult female relationships now she's an adult. She loves that she can rely on her step mum, she trusts her around sensible life advice (mortgages, wills etc) and knows she always has her best interest at heart. Her mum is lovely but a bit of a loose canon, she's not super helpful or sensible. So I also kind of hope for that with my stepdaughter and for you with yours.
Finally (sorry this is so bloody long), look at your relationship with your daughter, look at her upbringing, her successes and dreams and let yourself acknowledge that you do know what you're doing, don't let this other woman gaslight you into thinking you're a shit mum because a 9 year old doesn't like doing chores.