Sounds like an injector to me.
Thermostat is a possibility as well, although not a likely one IMO.
Maybe...high exh temps IMO, burning up the crap in the muffler if it still has one. Lots of causes for it.
I have seen diesels do this on heavy loads after prolonged light load running or sitting idling for a long time too.
I am NOT very familiar with diesels that have all the latest sensors related to emissions tho.
Some emission systems use a single temp sensor for both the gage/light and the emissions control, others use a separate temp sensor for the emissions part. You really need to get the temp gage working...it can tell you a lot.
If this were a gasoline engine of modern design, it would also use a MAT sensor..Manifold Ambient/Absolute Temperature.
It aids in fuel control and the software has a default setting if it doesn't recieve a real temperature reading from the sensor. I don't know if diesels use them or not. Default setting for most gasser systems was pretty low....mid 80s GM engines used -20F as default and idled really high because of it as the fuel control really poured the fuel in in an attempt to warm the engine up. You won't probably see high idles in the diesel as it uses a governor to limit rpms.
Just throwing some stuff out here....
You said the oil and coolant level are normal and neither drop nor rise. Do the oil smell/feel/paper towel test to see if the oil is diluted with diesel fuel. Diluted oil will feel differently between your fingers than regular oil, and will spread out on a paper towel much faster than good oil will, and you can usually smell the diesel in the oil.
You said it has been smoking for awhile. It's turbo I believe, and you may be getting oil past the turbo seals and into the intake, which will burn and I have seen old 'dumb' diesels spit out lots and lots of little chunks of carbon because of it.
The Deere guy mentioned the cold start and thermostat because a stuck open thermostat will keep the engine too cool and the fuel doesn't burn completely and ends up in the muffler. Modern thermostats are regulators, not full open or full closed like thermostats of days gone by. When the muffler gets hot, the unburned crap turns to carbon and gets thrown out the exhaust stack. Cold start advances the volume of fuel injected and can do the same thing.
I wish I knew more about these modern diesels but since I don't have one, I don't know much..