Flown with both suggested sets of SOPs, but definitively prefer the suggestion where FO starts.
It enables captain to monitor while simultaneously oversee pushback, especially when hand signals are required, use parking brake as required, signal/talk to ground crew while standard calls are being made by FO during process of starting engines who's sole responsibility at this stage is monitoring engine start...
Alternative, like PM/PF scenario, it ensures a good workload distribution when FO is PF. However when CP is PF he does ground crew, start engines, discuss delay codes, give hand signals if required, fuel levers and start switches, while FO only makes ONE SOP call (in present company): for Oil pressure...