Most likely the solenoid or wiring depending on the vehicle. A starter needs 2 things to operate. A current to close the circuit in the solenoid (small wire/trigger wire) and a power source to turn the starter itself (large battery cable). Battery cable goes directly to the starter from the positive battery terminal with an on/off switch in between (solenoid). The trigger wire (small wire) controls the solenoid by closing the circuit when there is power supplied to the small wire. If there is no power at the small wire then the solenoid must be stuck in the closed position unless the large battery cables to the solenoid are connected incorrectly.
if it is in fact the starter, it is probably engaging the flywheel. starter may be faulty.
The starter itself.. that would be unlikely... but a wire connected to the starter could on some vehicles.
why would my starter wire burn, going from starter to battery
No. A bad starter would cause the car not to start.
your clutch may not be fully engaging
The starter relay or the internal starter relay solenoid contact is closed either by 12 volts still applied to the relay or starter or the contacts have welded themselves together. Welded together sounds good if the starter is shorted and that would cause the cables to overheat also.
You need to replace the starter.
The starter is probably not aligned with the flywheel. The clearance between the starter and the ring gear needs to be checked.
The starter relay (solenoid) points are melted together.
bad starter or starter solenoid.
Starter selenoid no good
bad starter or bad connection to starter or bad ground on battery or starter