I have been having troubles with my thermostart so I understand how they work in my tractor and maybe your tractor works the same since they are both Yanmars. When I turn the key to the left the thermostart heats up and a red light turns on. This light turns off after maybe 15 seconds. While the light is on fuel from the little reservoir drips onto the heating coils, though the coils will stay energized as long as the key is turned to the left. But this is when using the stock thermostart. My stock thermostart burned out and so I bought a replacement. They work differently. When they heat up a valve opens from the heat and lets fuel drip onto the hot coil. So as long as they are hot fuel should be dripping on the coils. I ended up getting 3 thermostarts and none of them worked properly. They all get plenty hot but they don't let any fuel drip onto the coils so there is no flame in the intake manifold. I diagnosed this by removing the air intake hose from the intake manifold. On my tractor when this is done the thermostart is quite visible because it protrudes some into the air path into the intake manifold. So, even though my thermostart gets plenty hot it doesn't start a fire like it is supposed to. This means that the intake air won't get heated very much. So when it is cold outside my tractor takes a long time to start. When the air temperature is in the high 30s I have to crank the engine long enough that I worry about cooking the starter motor. And when the engine finally starts it puts out black smoke for a short period of time, just like yours. After the engine warms up it doesn't smoke. And when the weather is warm, say 60 degrees and above, the tractor starts right up. Maybe fuel is not getting to your thermostart. Pull the intake hose and take a look.
Eric