The other two black should be the charging from the alternator.
There's no voltage present on these until the alternator is standing still,
that's why i suggested to check the resistance towards the alternator,
just to make sure the function of these wires. (if they remain disconnected
while the engine is fired up that may kills the voltage regulator of the alternator)
If the solenoid doesn't "clunks" that might indicate burnt coil inside, but it's
not too common. The high current contacts inside the solenoid are became
defective more often, but in this case the "clunk" noise is present, only the
starter won't spin up.
So... now the thick black wire (with 13 Volt) is fixed under to the upper nut of the
solenoid, the braided one bolted under the lower nut and the "skinny" black wire
is connected as well, and still nothing?