Cranks slow with the key, fast at the relay
It does seem like there is a voltage drop (resistance).
The only question is where.
recheck using the internal resistance (voltage drop) method described above.
You should also check the ammeter during a normal start (with key). Should just be a small current flowing to ignition.
I saw the voltage regulator was smoking.
This makes me suspect something wrong in the 'ignition" circuit.
Is the regulator "seeing" battery voltage when the car is running?
Is the battery seeing the same voltage as the alternator is running at? Especially when the battery is recharging, but also when lights or other loads are on?
If not, the VR might have been full fielding for extended periods of time.
Use the handheld voltmeter and the dash ammeter together to figure out what's going on.