Check Engine Light confusion

i had my check engine light come on, on my commute to work i pullled over and had a look around the car(it was very dark) and noticed one of my tail lights was out i got back in the car and finished my commute i did notice that the car was not in limp mode so assumed that the CEL was on because of the tail light i changed it and the first start of the car after the CEL light was still on. but after restarting it a bit later the CEL light is off.

was the CEL light on because i had a tail light out.

i dont no if this will affect anything but it was -10oC on the drive in and my first restart after changing the bulb the second restart where the CEL light cleared it was -3oC

I think you have to start the car 3 times to clear the fault as long as it has been fixed.

i which case it must have cleared as its not on today either. the main question is will a tail light out light the CEL light