I don't know what that thing is in the photo.
All I know is the limit switch has several functions which would be consistent with the behavior you saw. It will: 1) turn the fan ON after the burner starts and the HX gets hot enough, 2) keep the fan ON after the burner is off to remove excess heat from the HX, 3) turn the fan OFF after the HX cools down too much, and 4) cut off the burner OFF if the HX gets dangerously hot.
You should get an HVAC tech who knows oil to diagnose things. You don't want to take chances with oil.