There isn't one name for them. There are a diverse bunch of therapies that are not psychoanalytical. For example, CBT, DBT, Family systems, humanistic, existential, Gestalt, and more.

My T is Family systems plus humanistic. He also does stuff like ego state therapy, lifespan integration, and EMDR. If he were to pick just one he would identify as, I think it would be Family Systems. We have lots of back and forth in our conversations. He is definitely not a blank mask. He values being authentic, genuine, congruent, etc. He is definitely there in the room with me. What's the point of therapy if the therapist doesn't want to be present?

I hope you find the sort of T that will work for you. You deserve it, mcl. At least you know what you don't want. Maybe you can screen them by phone and if they say they do the blank mask thing, just hang up the phone, quick!