The idea of "white knighting" is distasteful because people consider white knights to be motivated to protect women because they are women. Removing that aspect gets rid of the white knighting.

Ideally yes, but not necessarily in practice. I've been accused of white knighting before for engaging in behaviors that I not only would, but had, engaged in on behalf of men (exclusively in such cases, in fact, since I don't do much for women that I don't also do for men.)

Of course, people can only read observed behaviors, not intents or past actions, but I was hoping to get a wider response to my question, in the form of "this is the sort of response I would like to see," more than "responses motivated in this way are better than responses motivated in this other way." The example that you provided helps, but it's not always easy for a person to tell how their actions would appear to be motivated from the outside. It's not something I would personally be likely to say, but I can easily see someone responding with "that's a rude thing to say to a woman" simply because the thing they're responding to is rude to say to a woman, whereas to say it to a man would simply be bizarre