Nice videos and pictures, that helps. First I recommend reading through threads about "funky chicken" EDIC, etc theres lots of good stuff on here, but have to take time searching and reading.
Ill try to answer but I am by no means even close to a knowledgeable person on here, really search and read, the only things I know are from here and messing around with stuff.
Assumptions:
1. If the oil pressure switch was already unplugged, then the EDIC was not stopping the injection pump due to a low oil pressure signal. Therefore, oil pressure is not the problem.
2. The oil filter has oil and therefore is not contributing to an oil pressure issue. Ive had the wire not hooked up and then touch something and ground out and shut mine down before, (not common) and still good to verify oil pressure, gauge is cheap and easy.
3. The EDIC operates normally. The EDIC can appear to operate normally for days or weeks then randomly do its thing, warm or cold, you can also check the fuel control relay, in a BJ60 its under the dash along the passenger sidewall, reading on that will be in the EDIC threads.
4. Because the EDIC operates normally, my hard starts are not due to the EDIC malfunctioning. see above, also look at your starting and glow plug relays, grounds, fusible links, etc
5. If the oil pressure switch was unplugged, maybe it should be replaced with a new one anyway.
6. If the EDIC is operating normally and I am still having a hard time starting the engine when cold, then my glow plugs may be failing.
7. Because I am getting plenty of fuel and the truck runs normally once warm, it does not sound like a fuel problem.
8. Given assumptions 1-6, my glow plugs may be the culprit, even though my circuit test of them this morning showed that they were "working". You should verify they are getting the right amount of voltage (not sure on what this is with a 24v truck) and that it follows the superglow and afterglow (if you have these) usually the intial light up stage on the dash is the super glow then, thats off, and you start, then the afterglow is determined by the water temp sensor I believe, this can be done with the bus bar on. You will see higher voltage with the superglow then it will drop off to lower, then off completely. You can test the individual glow plugs as well by taking them out and using a multimeter, i believe its in the manual, and you can youtube it ( dont quote me on any of this!)
You can also have a leaky fuel line or injector, etc that will let a small amount of air in and only be a problem on start up but cause rough starting.
These are old vehicles with lots of rust and corrosion so these issues often pop up and can be difficult to track down
For your connectors I think 1 is also your sender unit, at least mine has a similar yellow black wire, and 2 is likely the low oil switch, or #3? 3 and 4 may go to your thermostat housing depending on your set up, in looking at my BJ60 I have only 4 wires coming off one wire bundle, 2 are the oil send and switch, and the other two go to the T stat housing, and they appear similar but not exact, Ive replaced a lot of my connections so I am not sure what the originals looked like.
I hope that helps some, again Im not even close to a expert on here.