Do parents "babysit" their kids??

HCBM says she won't "babysit" her biological daughter and I am at a complete loss. For context, SD13 lives with BD49 and me, SM48 60% of the time. BD has final decision making and this has been the CO since SD was 2. HCBM47 does not work and hasn't since SD was born. She lives off of welfare and baby daddy money for SD13 and BS18. Child support is based off of her ability to work 40 hours per week at minimum wage, and even though she only has her 40%, BD still pays around $300 per month. She refuses to pay for anything, school clothes, school supplies, school fees, pictures, extracurriculars, etc. We've gotten to the point where my spouse doesn't ever ask her to contribute anything, we just pay for whatever she needs. As an example, BD and I paid $150 for her track gear (I offered to split it with him) and HCBM bought her own self a track hoodie for $50 instead of offering to pat for what SD needed. That's just some context and it is what it is. I choose to help when I can because she's a straight A student, a really great kid, she's so dedicated to everything she does and she appreciates everything she has.

So, to my question at hand. SD has been taking English Riding lessons since October of 2022. She had been asking for a while and BD had a friend from High School who teaches he signed her up. She is one of those kids that eats, sleeps and breathes horses. She studies constantly and she is very committed. She loves it so much and it's been such a positive thing for her. Like I said, she is a straight A student, she's in the National Junior Honor Society, she works so hard at everything and never gives us any trouble. The last 6 months or so, her trainer had talked to us about her moving into half leasing a horse, which means I stead of her just having a 1 hour lesson on Sundays, she will be responsible for the horse 3 days a week. The lease fee includes all of the food and supplies. Leasing means that on her 3 days, she has to be at the barn 3 days a week this summer to groom and ride her, muck out her stall and do whatever else is needed. This is a step into potentially owning a horse someday, and goes beyond just taking lessons. We can choose the 3 days and she can even still be there on Sundays to help with the younger kids getting their horse ready for their lessons and if she does that, she can work off the cost of 2 of her lessons per month. In order to lease, the requirement is that she still has to have 2 paid lessons per month, but she can have 2 free lessons also on the days she's at the barn. As I said, we are paying the lease fee and the 2 paid lessons because that's just how it is. In my mind, it's absolutely worth it because it's so good and positive for her growth and development as a person. We came up with a sample schedule and one that would actually be closer for BM 's regular pickup from our house. I work from home and have arranged my Monday's to pick up my 2 year old granddaughter and we spend the afternoon/early evening with SD at the barn. We arranged for her mom's pick up day, for me to drive her to the barn (which means about an hour and a half away from my job) and there are 2 other trainers/adult women who will be there the whole time she is there. Then every other week, for her 3rd barn day, BD has been able to arrange for him to spend the day with her there. So here's the problem.....HCBM is not happy with the plan for the day she would pick her up at the barn because BD or I can't spend the day there. As I said, we have already planned for her to be there ONLY when 2 other women/trainers will be there too. So, since HCBM isn't happy with that option, BD suggested that since she doesn't work, she could go to the barn with her, I could still drive SD out there and BM can be there. The regular schedule for that day is we have her in the morning and take her to school, BM is supposed to pick up from school. That happens about 50% of the time, many days she decides she can't pick up.about 30 minutes before school is out and then we typically have her for a few hours at least while BM is deciding of she's picking her up from our house , but that's a whole other thing! Picking up at the barn is actually about 20 to 25 minutes closer for her. When BD suggested that she could spend a few hours on that day at the barn with her, she said "I'm not babysitting her on your day". Technically it's both parents day because BM has her overnight. I am seriously dumbfounded that she considers it babysitting her own daughter! Is this normal? I've never said that I was babysitting my bio kids (who are adults now), because they are my kids 100g of the time! Am I crazy for thinking it's weird??