The common connection (third wire) of the trigger contacts is grounded and in the original installation the ground was provided by running the third wire back to the controller with an in-out connection to ground. Theoretically you should be able to eliminate the third wire by just grounding the common point at the distributor if the ground connection at the controller is good. There are some merits to running an isolated common connection back to the controller; but, if 123 has not given you that option then there is nothing you can do about the absence of the third wire.
I have long since departed from D jet ownership; but, my recollection is that in addition to the shot of fuel occurring through the cold start injector the port injectors will open and close during cranking. You can check this by using ‘noid’ lights to confirm that the injectors are firing during cranking or actually pulling the injectors out of the holders to confirm fuel flow. Stick a short piece of clear vinyl tubing over the end of each injector and have it drain into a jar so you are not spraying fuel all over the place. Be aware that if you pull the injectors out of the holders to check for fuel delivery you will almost certainly have to replace the injector seals on the tips as they will not longer re seal after being disturbed. Be careful pulling the fuel rail and injectors out of the holders because you can kink the fuel rail if a couple injectors release while one or more remain stuck in their seals.
The cold start injector is controlled independent of the injection pulse width calculation so does not rely on the main injector control or the distributor triggers for operation. The cold start injector fires during cranking and once you switch from cranking to run the cold start injector stops flowing which could explain why your engine is dying when you transition off of cranking.
If your test of the injectors confirms no fuel delivery during cranking start by examining the plug connector for the distributor trigger points. The D jet wiring harness is noted for age / heat related failure. This includes crumbling of the insulation in the harness which may result in internal shorting. Any sign of insulation crumbling at the plug is cause to strip back the external jacket and check the wires for internal damage. You could have precipitated this damage by the mere act of unplugging and plugging back in the distributor plug disturbing the 50+ year old insulation. If the two connections from the 123 trigger contacts are not making good connection back to the D jet controller the system will be stone cold dead. The controller needs to get signals from both trigger contacts to deliver any fuel.
If the wiring looks OK, I would be inclined to put the original distributor back into the car and see if that restores operation. If it does then it is entirely possible that the 123 trigger contact circuit (which are probably just a pair of FETs) has failed. If it doesn’t then you should be looking for wiring harness issues.
If the car passes the fuel flow test at the injectors during cranking then you need to look someplace other than the trigger points for the no run problem.
Note that a trigger contact problem is not the only thing that can cause non delivery of fuel; but, since this all started with a 123 distributor installation this is a good place to start.