This thread made me laugh at myself. My first thought was... My T rarely gives me homework and at the beginning of therapy she never asked me what my goals for therapy were.
Then I realized.... that's because I beat her to the punch. I was soooo frustrated after our first 3 meetings and wasn't sure if she was the right T for me. So I sent her an email outlining what I thought my family's issues were, what I wanted to gain from therapy, what skills I was looking for in a therapist, and asked if these issues fell within her area of expertise and scope of practice. If not, I would appreciate any referral suggestions she might have. LOL! I just pulled this first email...I was so business...basically let's cut the crap lady...am I in the right place or not. LOL.
I never imagined how difficult establishing and maintaining a workable relationship with a therapist could be. At this point I would not want to start the whole process over again. I give you a lot of credit for doing this 4 times.
You mentioned that your previous T's were doing internships. Is this T doing an internship too? Are you in a situation that forces you to continue to see T's in training?
Either way... I think therapy is about what you want. Your goals are just that, your goals. They can be as detailed as you want or as open ended as you want. If I were to sit down and write goals today, they would be much more open ended.
It looks like you've got a good start on this homework assignment. If you want to throw in some therapy objectives do it.