Phantom:
My first suggestion to you is to take off the sheet metal cover on the welding generator & look on the inside of it. Lincoln used to paste a printed wiring diagram inside the generator cover.
If this diagram is not in place, call Lincoln Electric's tech support in Cleveland, OH. They are quite good about getting a copy of the manual to you, usually at no charge. I was in a jam with my own Lncoln engine driven welder ( an old "Weldanpower"), and they emailed me the wiring diagram as a PDF file, then followed up with a copy of the manual in the mail.
If you go to the Lincoln website and have the "code" on the machine, you may also be able to download a complete manual including wiring diagram.
Setting aside the lack of a manual, and admtting to no specific experience with an Onan engine or your machine:
The machine experienced an overheated engine. Afterwards, you note the machne wil "not crank". However, from reading your email, it sounds like the starter motor turns over the engine OK. My first guess is that one of the safeguards on the engine is keepng the ignition from firing during the start cycle. The engine may have a low oil pressure switch to shut down the engine in the event of a loss of oil pressure. Not sure about a high oil temp switch. I'd locate these two switches and check continuity accorss them with a multimeter.
I would also suggest you remove the low oil pressure switch from the tapping in the engine and screw in a mechanical pressure gauge with a good low-range scale (0-50 psi is plenty). Next, I do not know if your engine has electronic ignition vs. coil-&-points. Being only 9 yrs old, I am inclined to believe it has electronic igntion. For that reason, you should NOT try cranking the engine with the plug leads simply disconnected from the sparkplugs, or plugs hanging loosely on the plug wires, ungrounded. You need to either ground out the plug wires (if disconnected from the sparkplugs) or make sure to temporarily disconnect current to the ignition circuit before test-crankign the engine. With the gauge in place in the engine block, crank the engine. Note how much pressure builds and how long it takes to happen. I am unsure what the setpoint on a low oil pressure switch is, but I am guessing it is fairly low, like maybe 5 psi. See what oil pressure the engine builds cold while cranking.
Also test the low oil pressure switch on shop air to see if it opens when pressure is applied to it and recloses when pressure is bled off.
Not to cast doom and gloom, but as a worst-case scenario: the engine overheated and this may well have thinned the oil sufficiently to cause damage to the engine bearings with resulting low oil pressure. The damage was severe neough to keep from building oil pressure on cranking. I tend to think this is unlikely, since you mention changing the oil after the overheat incident. New oil cold, should be thick enough to allow the system to build a little oil pressure on cranking, even on a badly worn engine.
Or, the resulting overheat situation tripped a high temperature switch (if one is on this engine). This switch may not have reset.
These two safeguard switches are likely wired in series with the primary current to the ignition coil. It remains to be seen if they are preventing power from getting to the coil during starting.
My next guess is there is some kind of "start mode" vs. "run mode" on the engine control circuits. In the "start mode", the low oil pressure shutdown switch is bypassed to allow the engine to have spark while cranking up from 0 oil pressure. The start mode may use a relay to bypass the low oil pressure switch while the ignition switch is toggled to the "start" position. This same relay may also feed full battery voltage to the ignition coil(s) during starting. Normally, ignition coils are supplied with something less than 12 volts to the primary side. The 12 volts are lowered to about half that using a "ballast resistor". Some ignition coils have this resistor built into them, and some systems rely on external resistors. On older cars and some industrial engines, there was wiring in the starting circuit to bypass these resistors and put full battery voltage into the coils. The idea was the heavy current draw of the starting motor was going to pull down battery voltage, and a very weak spark would result. So, during starting, full battery voltage was supplied to the coil. Once the engine "caught", the ignition system was switched from "start mode" to "run mode". This would put the low oil pressure switch and the ballast resistors in series with the ignition coils. Any current going to the coils had to pass thru the safety shutdown switches.
I would start by NOT cranking the engine, put the igntion to the "run" position, and check voltage to the primary side of the coils. If current is getting to the coils, then the matter is a weak spark. If current is not getting to the coils, then the start checking continuity thru the wiring and components in the wiring on the primary side of the coils.
Once I had verified what was going on in the primary side wiring in the "run" position, I would repeat this in the "Start" mode. THis will mean crankign the engine. If no power is getting to the coils during cranking or "start mode", then I'd go back to the idea that one of the safeguards is keeping current from getting to the coils.
You mention a relay as clicking and some time delay, then seeing a weak yellowish spark. It is possible that Lincoln used some sort of time delay circuit (either a time delay relay or a solid state cricuit). This would automatically hold out the oil pressure shutdown (and any other shutdowns) to allow time for the engine to start and build oil pressure. It is possible this time delay circuit is not functioning as it should, not bypassing the low oil pressure shutdown switch. As soon as you let off the swtich when cranking the engine, there is no spark since the low oil pressure switch may be keping current from getting to the coil. After a time, the time delay relay (or timing circuit) runs out, and switches things to the "run mode". This is when you start seeing spark.
If the spark is weak, that is a whole 'nother story. I think you mentined putting new coils on the machine. Check them against the old coils to see if the new coils ahve internal resistance (the old coils may have used an external resistance). If the new coils have the internal resistance vs. external on the old, then lower primary voltage may be cuasing a weka spark. The overheating could have fried the plug wires, and then there is the condition of the cap and rotor. Condensers break down with age as well (if this system still uses them).
These are generalized guesses on my part. I ride old Airhead BMW motorcycles. These have air cooled "boxer" engines like the Onan, but that is where the similarities end. The Airhead engines still use the old coil and points ignition, with one set of points, one condenser and two coils. Not sure if the Onan engine has gone to electronic ignition since it is much newer than my Airhead bike engines ( 30 years old).
There is another method of checking to see where the problem lies. I can;t say I recommend it, but it is what some mechanics would do. That is to run a hot wire or jumper from the battery to the primary side of the coils, and try cranking the engine. If the engine starts and runs, then you know one of two things is going on:
-somethng was cutting out the current to the coils during the start cycle with normal wiring in place
-or, the spark during the starting cycle is too weak.
I would not try hot wiring until you have confirmed the engine has good oil pressure during cranking.
I tend to believe that with a multimeter, an oil pressure gauge, and some patience you should find the problem.
Joe Michaels