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I don't think "caring" defines a "relationship." I care about starving children in Africa, but I don't have a relationship with them. I care about the soldiers fighting in other countries, but I don't have a relationship with them. "Caring" is only one piece of relationships among people. Relationships involve two-way interaction and sharing, being there for each other, not just a focus on one-person. Relationships ebb and flow... between supporting people.

The caring from a T is similar to other professional caring.
I think Ts stating otherwise is actually more damaging because it gives clients an idea that they are more important than the reality of the relationship.