No, its not fuel left in the combustor at shutdown, (a properly functioning engine wouldn't have much).
To understand what I'm saying you have to know how an engine lights off. I'm going to be real general and basic.
There are usually 2 ignitors in the engine. 1 or 2 may be used for light-off, depending on engine type. These ignitors may be on opposite sides or side-by-side. At the front of the combustor there are several fuel nozzles (30 on CF680C). Since there is not 1 ignitor per nozzle, only some fuel is initially ignited. The start-up process depends on the propagation of the flame from the initial light-up.
As stated above a cold engine/fuel combination inhibits the propagation of the flame front. So, you have raw fuel dumping out the back until there is sufficient heat to ignite all the fuel coming from the nozzles.
After the engine reaches a self-sustaining speed, usually 50% N2
on big fans, the iginition system stops. The fuel burn sustains itself. Ignition is then turned on, automatically or manually depending on aircraft, during phases of flight as determined by the manufacturer or operator, i.e. icing, landing, heavy precip, etc.
Thinking it through, the reason some engines "smoke" more than others can be that 2 ignitors are used in start-up rather than just 1. But, that's a guess.
As a note, I have seen engines (notably RB211's installed on L1011s) smoke eben on the hottest days. This I believe is a function of how easily the flame front moves around the combustor to light all the fuel.