I felt that she was accusing me of being prejudice just because I'm white.
That could be the case, but you yourself acknowledged that she had been waiting longer than usual. Yes, there was a misunderstanding regarding why you didn't serve her in a timely fashion as you would others; you didn't realize she was waiting for you. The circumstances regarding this misunderstanding are a little murky. You say that you didn't see her, but also that you did see her but that you thought she was doing other things than waiting for you. At some point something caused you to realize she was waiting for service. What was that and how was that different from the previous 20 minutes?
I'm not accusing you of anything; I'm just saying the circumstances provide a lot of room for confusion. In that case, the best thing is to de-escalate the situation. One option is to offer the service at a discount or for free, which you did. That was a good alternative. That the woman refused your offer could suggest that she felt a little bit guilty, like maybe she was over reacting but that she was having trouble stopping herself. Once an emotional suspicion begins in the spirit it can be difficult to slow it down even when a rational explanation is offered. Also, you can say something like, "Sorry, I didn't realize you were waiting for me as I thought you were doing other things; how can I help you"?
I think you did something like that, but you also attached this "I forgive you for thinking badly of me" thing to it which sounds like a respectable way of hitting back. This kind of thing is sometimes referred to as being passive-aggressive. It sounds fine on the surface; I forgive you, but I think if she was saying something similar to you (i.e. I forgive you for treating me differently) you would recognize the spirit behind it was not one of genuine reconciliation. In other words, it's a way of letting them know, without actually saying it, that they've done something which needs to be forgiven. Maybe they have done something that needs forgiveness, but hiding the accusation behind the pretense of offering forgiveness even though they've not recognized anything that they need to be forgiven for communicates to them that they're being falsely accused. They themselves may not consciously recognize this intellectually, but they will discern it in their spirit; humans are pretty good at picking up on hidden digs and accusations. I don't know if you've ever experienced it before (I know I have) where someone does this to me; they say, "I forgive you for that thing you did wrong" though I don't agree I did the wrong thing, so in retaliation I'll say, "oh yeah? Well I forgive
you!" and then the response is, "No, I forgive
YOU"!
It may also be helpful to recognize that under the circumstances
she believes she has good reason to suspect you. If you can try to empathize with why she'd be so sensitive, then it may help you to realize that she's not really upset with
you personally, but rather that she's struggling to process what she believes to be part of systematic racism occurring in the U.S. at the moment.
It would be better if she did not hold a grudge against you for what you've already explained was just a misunderstanding; it would be better if she took the lower seat and accepted your explanation in good faith, but it would also be good for you to recognize why she would be reluctant to believe you. In that case you could take the lower seat by perhaps apologizing again for the misunderstanding and just accepting that if she wants to believe the worst about you, then there's nothing you can do to force her to think differently. I realize that doesn't feel very nice but I think it can be an important part of humility. Sometimes we just have to accept that no matter how good and right our explanations and motivations are, some people simply will not care. They're too hurt or upset or emotionally involved to slow down and think carefully about what's happening and why it's happening.
The best way to combat that kind of emotional snowballing is to not allow ourselves to become part of it even if the situation genuinely makes us feel hurt. Setting those feelings aside and instead deliberately insisting on handling the situation rationally will be the best way of working through the issue.
Good luck.