I expect mine to care about me and not to do anything to hurt me intentionally.
I don't expect constant or frivolous between-session contact, but I like to know that he'll be there when I really need him.
For example,
Also, the last time we got in a fight during session and I emailed him the next day asking him to help me stop therapy and told him I was crying, he responded within two hours.
I guess what I really care about isn't so much his response time but knowing that he cares about me. And for some reason I equate his responsiveness to his caring about me. Maybe that's silly, I don't know.