I made a transition from taekwondo to Shotokan, and years later from Shotokan to Tang Soo Do (well, Chun Kuk Do). Each time, it was just like you describe: many similarities, but not identical. Each time, I wore a white belt and progressed through their ranks, though the instructor will typically do this at an accelerated rate because you're not really starting at zero. I didn't hide that I had prior experience, but I didn't go around pounding my chest about it either. When asked to do a technique, I would do it as I knew how to do it, and if there are corrections to how they do it at that school, I would accept the feedback and try to do it as they do it at that school. If I had questions about "why this way and not the other way," I would ask those questions respectfully after class, not interrupt the class with them, because then it comes off as undermining the style.