These are all similar, and different leagues have different meaning for some terms.  But the most common definitions are :

Table Run : During your first attempt; you get all your balls in while some balls were already sunk by the opponent.

ERO : While all balls are on the table you get all your balls in on your first attempt.

Eight ball (or Earned) Run Out

Break and Run : The player who breaks gets all his balls in. The game is over. This is better than an ERO.

You can decide which feats you enable on your scoresheet and which feats you show in your stats and reports.