With the key in the START position there should be 12 volts from the ignition switch. The START circuit does not go through the ballast resistor. The system uses the full 12 volts to start the car.
With the key in the RUN or ON position there should be 12 volts from the ignition switch. The RUN circuit runs through the ballast resistor which causes the power to drop. So there is 12 volts into the resistor but less coming out.
There is nothing in the ignition switch to change the voltage running through the switch. It's 12 volts in and 12 volts out. It's the ballast resistor that drops the power.
For more detail, when the key is in the START position power runs through a Brown wire to the resistor. At the resistor a Dark Blue wire should be attached with the Brown. Power runs through the Dark Blue wire to the coil, skipping the resistor. A full 12 volts from the switch to the coil.
In the RUN position, power runs through a Dark Blue wire to the resistor - opposite end of the resistor to where the Brown wire attaches. Power runs through the resistor and then via the Dark Blue wire to the coil. It's a full 12 volts to the resistor, but reduced power from the resistor to coil.
From the above you should understand why, when the ballast resistor dies, the engine will turn over and sound like it is running in START, but when you release the key and the ignition switch goes to the RUN position, everything dies.
A second Dark Blue wire is attached to the resistor with the Dark Blue wire for the RUN position which runs to the IGN connector on the Alternator Regulator. Has nothing to do with what we are talking about.
All correct but the function of the ballast resistor is not really in question. From the OPs description, he has correctly wired his ECU to the RUN side of the BR, but he says that during cranking his ECU is not getting enough voltage to operate. He said he's using a Mancini ignition conversion kit, which is their version of the Mopar orange box kit.
Voltage drop through the resistor is proportional to the current draw (Ohm's law). I don't know how much current a Mopar ECU draws but it's probably not much, and the BR is only about 1-ohm when cold, so the Mopar ECU should run fine through the BR during cranking. If the battery voltage stays at 12V during cranking, the ECU draws 2A and the BR is 1-ohm, you should see 10V at the ECU tap point during cranking.) That's why we can connect it there, even though the RUN wire doesn't get power during cranking so the ECU gets its power through the BR instead.
If your multimeter is sensitive enough, you could directly measure the resistance of the BR to confirm. Set the meter to the lowest resistance setting. Touch the two probe leads together. If the measured resistance value is not zero, your meter has some internal resistance which will affect the reading. Subtract that value from what you measure when probing across the BR terminals to get a more accurate reading. Disconnect all wires from the BR and measure across the BR terminals. Should read around 1 - 1.5 ohms. If higher, that could be the problem, or at least part of the problem..
The OP did not state where he was putting the negative probe lead while measuring voltage at the BR wires. He also didn't state whether he had pulled the plug on the starter relay so the starter was not actually running during measurements. (I presume he did not.) These may affect the readings greatly.
Here is where I would continue diagnosis: First make sure your battery reads 12V with the key off. If not, attach a bench charger to charge it before continuing. (Don't test with the charger still connected. it will affect the measurements.)
Next you'll need a friend in the car to turn the key. Make sure loads such as heater blower and headlights are off. Pull the connector off the starter relay on the firewall so the starter does not run during START. Pull the leads off the BR and test them again in RUN and START positions. If the START lead measures low voltage with the key in START, suspect the ignition switch is bad. If you get close to full battery voltage, the ignition switch, bulkhead connector and related wiring are not the issue.
Reattach the connector to the starter relay. Disconnect the ECU so the car doesn't accidentally start during testing. While cranking check:
- voltage between battery + and - terminals. (Measure right on the battery posts, not on the connectors.
- voltage between battery + and the engine block. (The alternator case is usually a good place to find a solid connection.)
- voltage between battery + and the firewall, preferably on one of the screws that ground the ECU to the firewall.
If all of these measurements fall to 6~7V during cranking, I would suspect a bad battery. If the voltage to the block ground is significantly lower, suspect bad terminal connections on either cable or the negative battery cable from battery to block. If your negative cable is original it probably needs replacement. If only the voltage to the firewall ground is low, suspect the ground connection between block and firewall at the back of the right cylinder head, or the ground to the ECU itself.
If none of these are bad, we'd be back to suspecting the ignition switch or related wiring, but these should have been ruled-out already.