I volunteer with kids, and if one of them came up to me and said, "Do you care about me?" I would never dream of saying, "I care about all my students," to that kid. I would instead say something like, "Of course I care about you. I also care about everyone in your class." Maybe it's a really tiny arbitrary thing, but to me there was just something about that particular wording that felt really wrong to me.

I think this is the crux of it. The difference is you're not a child, and your interaction is within a therapy relationship. But your lingering feeling of dissatisfaction about her response is very much as you would imagine the child in your example feeling. That's something to explore.