Hm, im not American. And what you describe in your first post sounds like a normal t-client relationship.
I would never expect outside contact, no matter hoe messed up I leave the office. If I met my therapist somewhere, if smile at her to acknowledge her, but id leave it up to her to decide if she would say something or choose to ignire me. Id love for her to interact. But if not, I'd tell myself its fine. Why would I expect any text? If its important, I'd mention it in the next session.

So, having said this, I niw got a very awesome therapist who even tells me things about herself if I ask. I love her for it, and I am starving for a deeper connection. Id live to be frienda with her, and for the first time, I am fully opening up in therapy. I want her to know me. I trust her to handle my darkest issues.
I love the connection, and I appreciate its deeper than in my home country. Not sure if all of it is cultural. I'm sure its a matter of therapist personality and character, too.

I kind of understand that my attachment is strong. I understand she is my therapist, and I may never get from her what I am starving for. On the other hand, I know she is helping me more than anyone ever has, and that's a good thing. In the end, all this may tell ir teach me more about my relationships with people in general. I hardly hace my friends, but even if trabsference is happening herw, I have decided to embrace it and continue with a great therapy.

Its no reason for me to doubt therapy being the right way.
Talk to him about his boundaries. Knowing may help you understand and change what you expect.