Always runs…; You are correct.
Sam2; Your answer is partially right. A presiding rabbi is not critical for the marriage to be Halachically valid. But in the case of Reform marriage, the entire process is flawed from a Halachic standpoint. But it takes such a strong “Posek” as R. Moshe Feinstein z”l to legally invalidate the first (Reform) marriage when there is a need of wide-ranging impact. It is not for us lay people. Where the couple stays together, there is absolutely no change in the status of the children before, or after they remarried in an Orthodox ceremony.