If you have someone else to help you test.... at its coldest try a two second spray of starter fluid into the intake tube before the air cleaner just after you engage the starter.
If that really helps then your injectors are certainly suspect. It could also be the condition of your precombustion chamber innards. That pintle helps disperse the fuel spray to the 7 tiny radial holes which then go to the bore... this is a much ignored but important part of a good running diesel engine... these types of things show up at the margin of the working envelope... IE... cold weather.
Without close inspection of the electrical diagrams of your specific model I do not know if the light and plugs can function fine without each other being affected.