Yes, I understand you didn’t mean it, but that’s how words on a screen can be misinterpreted. Apologies are negotiations. You give up some ground. They then have the opportunity to voluntarily give up some ground if they wish. By demanding an apology, you’ve given nothing and expected them to give you everything in return. In an argument that was alright heated in their head! Was never going to work.
It’s not necessarily about what you have to apologise for, more about what you wish to achieve. If you’ve offended someone for whatever reason - even if you can’t see any remote reason why - showing some contrition calms the situation down. If they really have attempted to gaslight you, it will be apparent to everyone else what they’ve done and you did your best to keep it reasonable. By questioning and reaffirming your position, in her head it validates her position.
You explained yourself well, but it reaffirmed (and possibly even interpreted as disregarded) the aspect she found offensive. She may have wilfully misinterpreted each step, but the way you responded added fuel to the fire. Notice when she started the argument about the use of language, your first response was to say it was when you were 6 - an entirely different part of the statement to the bit she was complaining about. In her head, that could be a clear strawman to distract from the argument. The way she reasserted her position was blunt and sarcastic, when she could have just said that part wasn’t the issue. You obviously responded more to the vitriolic sarcasm rather than the content. From there it spiralled out.
Again I can see your intentions were true (and probably hers too!), but I can also see the way an angry stranger on the screen could interpret them. I don’t think it’s reasonable to act as offended as she did over an issue that is so obviously contentious, but there were several opportunities for both sides not to take it so personally and quell the argument.