It sounds like the bad starter may have left some metal pieces in the flywheel housing. If there is an access port, open, check and remove any garbage. Also check the flywheel for broken teeth caused by the bad starter(That would be very bad$$).
Next, if possible have your battery load tested and recharged.
Before you install the new starter, disconnect the negative battery cable. Then clean both battery terminals and the connections at the solenoid and starter and the connection where the negative cable connects to the frame. Dirty or loose connections act just like a bad battery. Connect the negative cable last when you are reconnecting.
After you install the new starter, turn on the headlights and try to crank the engine. If the lights go dim, the battery needs recharge or replacement. If the lights don't go dim and you hear clicking, and you have cleaned all connections, the starter solenoid may be bad.
If the starter works but you hear grinding noise instead of proper starting noise, most likely is that flywheel teeth are damaged. Or starter could be misaligned. Remove starter and use a mirror to inspect flywheel teeth.