Forget anything about the cluster for right now. OR THE COLORs, or the connectors. lets just follow some wires
You must have some positive all the time larger gauge wire running from the battery to the BAT. Always hot if battery is connected.
You must have a wire of some type running from starter solenoid to the SOL.
You must have a wire to the ignition coil (or computer and all the stuff on the LS engine) from ING 2. (NOT ING1)
This will get it cranking and a running engine and should turn off when key is moved to ACC ( or stop)
You then have some type of wire running from ING1 to gauges.
Acc can have anything you want running while your just sitting there but not running the engine like radio.
my guess without instructions is that the adapter cable you have goes on that middle switch of the three. The white connector goes on left three posts (SOL), ING1, ING2
The pink wire end jumpers over to ACC. But this may also keep the truck running until you turn key to stop. Some folks want that.
and the black plug goes to the harness.
PLEASE validate this setup by following the wires to where they are supposed to go above.
But until you validate where those wires go from and to above you are not going to know its right because if one of those wires is not going where its supposed to its not going to run or gauges work.