Don't read this harshly, but I'm not going to say what I think you want to hear; I do understand how painful this sort of moment in therapy can be, and you can get through it and be stronger for it

I don't see any evidence that your discomfort with her boundaries is a result of her problems. You've said she's been a good T and that you've made progress and have a good relationship.

When she refuses to tell me she cares about me even after I told her I need to hear it, this makes me doubt all of her caring gestures, and it makes me feel like maybe I can’t trust her or like I got “tricked”.

When she said she cares about all of her clients, instead of feeling comforted by that--because it is evidence of her caring nature--you feel it as a competition: that if she cares for others, then she can't care enough about you. Do you see the difference? The first is an adult perception; the second is how a child developmentally views a parent. It's what people call "sibling rivalry." It's a stage.

It's not healthy or functional to insist on "proof" of caring from someone. There's nothing wrong with the feeling of need--and your T validated that--but to then try to coerce or compel a demonstration from someone else is a violation of that person's right to their own expression of emotion (or their choice to not express an emotion). It's also fruitless because the demonstrations are never enough--they never fill the need. That's why she's holding the line and refusing to be manipulated.

It's a fine line between asking for what you need, which is a mature action, coupled with a willingness to explore what the need means, ending with an acceptance that the other person has the right and responsibility to respond as they see fit;-- and demanding a particular response, not examining the need, and then projecting one's frustration onto the other person and using it as a rationale for why they're wrong.

She's doing well by you. The hard work of therapy can be coming to terms with such needs in ourselves and learning how to meet them within ourselves in a healthy way.