Now, I have to admit, that I am only two lessons in, but I feel a little uneasy about it. I am not sure whether this is a case of learning EXACTLY what I asked for (cords/ improv/ playing from lead sheets) and I just don't know it yet, but is this something I could expect (based on my initial spec)? How long would you give it before deciding Whether I need a different tutor? Should I be trialling someone different NOW (there is another tutor in my area who does Jazz).
I think you are premature in your feelings.
First, it takes a while for a teacher to understand a student, which includes what they know, what they don't know, how fast they learn, what style of learning is best, will they practice, will they practice correctly, etc, etc.
Two weeks is early in the process.
Second, students such as yourself present a somewhat difficult task for a teacher, for at least two reasons.
The first is you are not a brand-new student, so the teacher does not have the luxury of starting off at the beginning. Instead, he/she must play the part of a detective, and discover what you know, and what you don't, and try to fix bad habits all the while keeping you interested long enough to have fruitful progress so you don't quit before the cake is ready to come out of the oven.
Second, you are wanting to:
move away from classical leanings and more into Jazz, contemporary music, including and ultimately improv.
This means that the teacher does not have the method books to teach you the "traditional" classical way.
Thus, for music other than classical, and especially for improv, it is a "lets explore this together" journey rather than teach step by step from a standardized method book.
I have been down this path with numerous students, and it has taken as much as several months before we settle in, and I learn the person's weakness, try to fill in the gaps, and then finally move on to forward motion in the relatively uncharted waters (compared to classical) of pop, blues (I don't teach modern Jazz,) lead sheets, and improv.
What he did with Watermelon Man with you is exactly what I would do to see where you are, and how you respond. Plus, that choice of music fits in exactly with what you asked for. I don't get your problem.
I suggest that you work real hard, and see where you are in three months, if you are making decent progress.