The problem as described is the "OE" Solenoid.
If the O.E> is still out of the Car obtain an inexpensive (Affordable) replacement Solenoid and swap out the "REBUILT" O.E. and re=bench test the assembly.
My thinking is if Someone Rebuilt the O.E. Solenoid they may have miss wired the Pull in power wiring and when the Plunger is energized the only way to de-energize the coil is to remove the Hot power to the Main stud (Contact).
Photo of simple unloaded bench test, Do not allow starter to spin longer than about 3 sec.
Ed had a "great idea", Spring forces the contact open breaking the contact. Bet he is correct. Oh,,,,,,,,,,spring is in,,,Hum
Jim