Since you can start the engine with the key in RUN by jumping the solenoid, you can eliminate everything that has to do with the EFI. It sounds like you are losing power to the ignition system and/or the EEC whenever you go to the START position. The ignition switches in most cars cut power to all the accessories whenever the switch is in the START position. For old cars, the ignition system gets power form the solenoid's "I" terminal while cranking. For newer cars, a very complicated ignition switch handles all of this.
I would troubleshoot as follows:
1. Pull the small red/blue wire off the "S" terminal of the solenoid so that you don't crank the engine while you are performing the next steps.
2. Disconnect the 60 pin connector from the EEC.
3. At pin 1, make sure you have +12V at all times - this is a constant +12V input to the EEC. Check it with the ignition switch in OFF, RUN, & START.
4. At pins 37 & 57, perform the same checks for +12V. You should get 0V in the OFF position & +12V in the RUN and START positions. (My guess is that you are not getting +12V at these points in the START position).
5. If you are getting +12V in the RUN position but not in the START position, your EEC power relay is dropping out when you go to START. If this is the case, you need to supply power to the EEC power relay while cranking. I can give you some ideas on how to do this if this turns out to be the problem.
6. If you are seeing +12V in the START position, perform the same checks at the positive feed to your ignition coil.
7. If everything checks out fine, post your results here and I'll dig further into schematics.