Is it OK if my Essay Doesn't Have a Defined Thesis?

Since it's a personal essay, not an argumentative one, it is probably OK not to have a thesis per se since you're not really trying to prove an argument. You'll have something kind of like a thesis in that your story is supposed to say something about you, and it should be clear by the end what that is, but it doesn't necessarily need to be laid out in black and white in the first paragraph. As long as it's internally coherent with a beginning, a middle and an end, it probably reads a lot better as a story than in a dry, formulaic essay format (not that argumentative essays have to be dry and formulaic either, but you get the idea).

If you are applying to an honors college, admissions will be looking for an above average essay. If your essay flows really well, then you have a good start but college admission is really competitive these days and a good essay isn't good enough. It has to be perfect. I would try to add a thesis.