Nowhere is the evidence for a direct line from association football to association (field hockey) to ice hockey more obvious than the text of the offside rule.
Here is the offside rule from the original Montreal ice hockey rules:
2. When a player hits the ball, any one of the same side who at such moment of hitting is nearer to the opponents' goal line is out of play, and may not touch the ball himself, or in any way whatever prevent any other player from doing so, until the ball has been played. A player must always be on his own side of the ball.
Here is the Hockey Association version of the offside rule. Remember that it was noted when the first Montreal rules were published that they were based on the Hockey Association code, which is obvious when you read them:
6. When a player hits the ball, any one of the same side who at such moment of hitting is nearer to the opponents' goal line is out of play, and may not touch the ball himself, nor in any way whatever prevent any other player from doing so, until the ball has been played, unless there are at least three of his opponents nearer their own goal-line; but no player is out of play when the ball is hit from the goal-line.
And finally, the 1863 Association Football rule:
6. When a player has kicked the ball, any one of the same side who is nearer to the opponent's goal line is out of play, and may not touch the ball himself, nor in any way whatever prevent any other player from doing so, until he is in play; but no player is out of play when the ball is kicked off from behind the goal line.
The origin of the ice hockey offside rule is clear as day. Now, if we look at the offside rules from the 1871 rugby code, we see that while they have some slight similarity (in that it's an onside game), they are presented very differently, and also have some important differences.
22. Every player is onside but is put off side if he enters a scrummage from his opponents' side or being in a scrummage gets in front of the ball, or when the ball has been kicked, touched on or is being run with by any of his own side behind him (ie between himself and his own goal line).
23. Every player when offside is out of the game and shall not touch the ball in any case whatever, either in or our of touch or goal, or in any way interrupt or obstruct any player, until he is again on side.
24. A player being offside is put on side when the ball has been run five yards with or kicked by or has touched the dress or person of any players of the opposite side or when one of his own side has run in front of him.
25. When a player has the ball none of his opponents who at the time are offside may commence or attempt to run, tackle or otherwise interrupt such player until he has run five yards.
26. Throwing back. It is lawful for any plaeyr who has the ball to throw it back towards his own goal, or to pass it back to any player of his own side who is at the time behind him in accordance with the rules of on side.
In rugby, you could only pass the ball to a player who is behind you. In hockey, you could pass the puck to a player who is not in front of you at the time the pass is made. Rugby only allowed back-passes, hockey allowed laterals, and indeed allowed the puck to travel forward on the pass so long as the recipient was not ahead of you when the pass was made. The wording is completely different, and the effect is substantial.
Now, there may be some rugby influence in the OHA offside rule. Rule 24 above has some wording similar to the original 1891 OHA offside rule, which was worded differently than the Montreal version. But this was 16 years later, and has nothing to do with the original ice hockey rules, which eventually subsumed the OHA rules anyway.