The fact that the starter solenoid was smoking is a clue!
Batteries have the ability to produce thousands of amps when shorted. The alternator does not produce any current until the engine is up to speed, definitely not occurring while starting. This makes the alternator nothing but a drag until the engine has started, all the power is coming from the battery.
The 10 volt reading simply meant that you had an excessive load on the battery. The voltage drop could have been in the battery or the battery cables, I “volt” for the battery cables (sorry, I couldn’t help myself).
The alternator is probably fine. Check the battery voltage once the engine is started, if it is up to 14 volts + then it’s OK.