Whoever installed the new starter may have gotten the wiring a bit mixed up. On the mini-starters, the big cable running to the starter is supposed to be hot all the time. The little wire is switched from fender relay. The old, big one-wire starters had the big cable on the switched side of that relay so the big cable was hot only when you turn the ignition switch to 'start' and the fender relay engages. If you hook the mini-starter up that way, the symptom is that the starter keeps running after you try to start the car.
Michael Yount - Charlotte, NC - 82 Volvo 242 - 6.2L; '15 Fit; '16 CrossTrek