But within OPs house, I do feel it's fair enough for her to factually state that the steps/halves are steps/halves.
She isn't factually stating it though. She's constantly correcting, which is different.
From the daughter's lived experience, she has siblings who live in daddy's house who all just get to be siblings and she is separated from that.
Physically, it can't be helped because they live in diffeent houses.
Emotionally, though, this gap is bridged for the daughter by referring to them as her siblings. She knows they are technically half siblings but that doesn't matter to her because they are still her siblings.
Dropping the half makes them feel closer to her and increases her sense of belonging and family. This will all benefit the daughter.
None of this is a threat to the OP. However, trying to drive a wedge in between the relationship by constantly correcting her will build a resentment within her daughter as she grows and may well leave her with issues around her own sense of self and self worth which would need addressing in the future and that could well impact on the OP's relationship with her daughter in the future. All l to appease the OP's ego, which sounds harsh but that's what it's about. Not whether the daughter's description is biologically factual or not. None of this will benefit the daughter.
Her feelings around this are something the OP.needs to manage and address herself and not by controlling her daughter's language.