I did not view the clip. In my mind, the differences are:
門:Normally are niche and smaller scale groups, do not have large-scale recruitment of apprentices. Might be associated with security businesses like 鏢局。E.g. 快刀門
派:Most generic grouping. Could be applied to big setup or OMO. Could be established, highly-skilled or mediocre. E.g. 武當派, 少林派, 古墓派, 崑崙派.
幫:Normally used for low-SES groups (low-SES here is based on perceptions, not on actual wealth). E.g. 鐵掌幫, 丐幫.
教:Groups where doctrine or some religious systems were in place. E.g. 全真教, 明教
會:Rebels. E.g. 紅花會