I suspect there's a number of PPs not reading OPs posts or updates properly. OP did say "no sorry" to child when asked. A child had been bullying DD all week, literally not being kind, she made DD cry every day by her treatment of her and then also badgeting for an invite.
Child was not satisfied with that and got loud in her demanding, so OP in the end used exactly the phrase therapists and teachers use when asked why "you haven't been very kind to X (DD)" . Specific, brief, clear, factual, gentle as possible, and enough to stop further bullying of her DD and even OP, by this child who wasn't taking simple "no, sorry" for an answer. (DD would likely have been there at the time further watching the bullying attempts continue by this child into her party!)
Why the other mum did not redirect her child is the problem- but there may be reasons, one of which may include lazy parenting. (When my child comes out of school, I'm watching for her and her siblings, not letting her merrily accost people, raise her voice and demand invites to their parties. 😮)
The child cried because she was told no and was unable to bully DD or DDs mum into giving her an invite. That's sad, as no one likes it when a child cried, but it was caused by child's own parenting so far, not OP. OP has not indicated in anyway that she is rejoicing that the child cried at all - she posted here as it shocked her and she felt bad (even tho not her fault)- nor are YANBU PPs not saying "she deserved it" 🙄. Some PPs who disagree have exaggorated and changed the tone of & phrase that OP actually said. Why claim OP told she was "mean" or "unkind" as a person, when OP didn't do that at all. Why not read the posts properly? It doesn't help the discussion.
Those saying how awful it was as she is just a 6 year old child (same as OP's DD!) (but one who is also not behaving kindly for whatever reason) - may possibly (and we can't be sure, but I'd take that bet!) be the parents whose children visibly go around bullying others without their parents stepping in.
It's done and dusted now. Enough PPs have said, no that was fine OP, it was a difficult situation that she dealt with mildly but in a clear way.
Hopefully the other child might have enough parental support to help her see that maybe if she treated others kindly she'd be invited more. That's not OPs issue to sort out.