Orcish Bowmaster didn't trigger multiple times when opponent used the one ring to draw multiple cards in game 2. I believe this may be caused by phyrexian tower sacrifice of bowmaster that was reversed.
Can confirm that if you sac Orcish Bowmasters to Phyrexian Tower, then undo that action, that the Orcish Bowmasters will no longer trigger on card draw. I also blinked Orcish Bowmasters with Displacer Kitten, and then the Orcish Bowmasters functioned as expected again.