Yes, a starter can short internally. And a relay can short internally or externally, in this case more likely externally.

Check your firewall mounted solenoid and see if it heats up excessively. It should get warm, but not scalding hot, when you tap the key (yeah, I know, it'll kill the battery). So before you do the power test, look very carefully at the heavy gauge wire between the firewall mounted solenoid and the starter, and find the spot where the insulation is through and it is arcing to the block or exhaust or subframe.