If you're accelerating towards an orbital object you are by definition not in the same orbit as the target object. If you were you would be moving at the exact same speed. This effect makes getting yourself close enough for docking all sorts of tricky if you don't know what you're doing. Once you get close enough (like in the linked simulator) you can mostly ignore the problem since the effects are small given the relative speeds involved.

I can heartily recommend messing around with orbital maneuvers in KSP to get a feel for the problem. It's not exactly intuitive.