vroooom ptssssh: To be honest you would have to leave the fuel pump running a good couple of minutes before starting to majorly affect the cold start, sounds more like your car is miss-firing when you first start it (some do this), basically the auxiliary air device bypasses the throttle housing and allows a little extra air flow to raise the idle speed at cold, so the worst that could ever really happen is it would keep stalling and/or idle very low and lumpy. Letting it prime for a second is fine, the relays allow priming for 1 second to ensure a positive start, personally i would just get yourself a relay, safer that way anyway.
As for the green relay, its not a series 1 relay no

The series 1 relays are quite big actually, i dont know why theyre different to be fair, but i expect yours has been changed as the original relay packed up and theyre very hard to get hold of, and a lot of £££ i expect.