Idk about most but I use screen capture software to record SDT loading. I used to use Fraps but I no longer have access to it. I recommend OBS. Or you could use a smart phone to record the errors popping up.
SDT does have a log tho, but I don't think it records errors. Press I for the modding menu, then under Custom Dialogue click Edit. There's a button in this menu labeled Log. It shows all the variables and dialogue lines being loaded.