I am not a mechanic, and know nothing about your engine, but before you let one change fuel distributor and injectors ask what about cold start is different than warm run or warm start for those parts.
I know of no part of the fuel distributor used specifically for cold start. Leaking injectors cause warm start problems. Bad injectors should cause cold or warm problems.
There might be parts on your engine used for cold start such as the Thermo-time switch, Cold-start injecter, Auxiliary air valve etc.
Turning the ignition on turns on the fuel pump for a couple of second to build fuel presssure. Since that helps cold start you could have fuel delivery problem. For example if the fuel accumulator is not fuctioning after the car sits you could get vapor lock.
I think if you deal with a good mechanic, he should have the knowledge and test equipment to check all engine systems, and give you definate reasons to replace any part!