As far as the approach, it is all about what you are trying to get out of it. I can tell you as someone who teaches private students as well, I very much appreciate when students come with questions and show that they want to learn. You really can't put all the hope in the world on one instructor to realign your universe. If you come with questions, that really motivates the lesson and makes the instructor function much better instead of them conjuring up whatever it is THEY think you need to know. Tell them what YOU want to learn.
To do that, mix the two worlds together. At your next lesson, take a lesson or concept from GT that you are learning and find something that interests yet puzzles you. Be realistic and be able to accurately assess your ability and find something that you understand, but want some further clarification that only a private instructor can give you. That way, both of your worlds are helping each other and you aren't completely lost in one way or the other. Whenever I take lessons, I usually bring to the table concepts and questions that pertain to gigs I am doing or musical situations that I am not so confident about. That way, both worlds work together.
Do what you can to make the instructor work for you, and don't go in without something to talk about or work on. If you do that and they steer you into something you don't want to do, than find another instructor. I firmly believe it is about giving the student what they want.
Hope this helps. :D
Douglas Showalter