In competition, it sometimes happens that you are left with an easy two or three balls or an easy break shot, or you are left with 15 hangers. In any of these cases the run would count. So in any case it seems that wherever a run really begins is going to be somewhat arbitrary. But I say you at least have to do some straight pool-like work by breaking apart the rack. Spreading out 15 open balls in a really easy setup is really like giving yourself free balls for a run (assuming you count them), as you don't have to employ any straight pool strategy to get them down. You could set them up as easy as you want to. But starting with a break shot, you do have to work through them in a straight-pool-like fashion. Although any of these situations can come up in competition, you are starting out with a straight pool shot to begin the run.
So I say in PRACTICE, a proper run starts with a ball and break shot, however you get there. On the other hand, if you miss a ball in a run and then continue shooting to start another run (just so long as the miss was not intended to give you a wide open table for another run), it seems legitimate to count that as well. Really I think the only run that is not legitimate would be giving yourself a wide open table on purpose to give yourself free balls.
So really none of the options in the poll seem to fit this.