The FAQ says:
Q05. Who is considered to be responsible of the elimination of a player with cards like Indians! and Gatling?
A. Unlike the Dynamite, the player who played the Gatling or the Indians! is considered to be responsible of the elimination of a player. All special actions due to this elimination apply to him (for example, if he just killed an Outlaw this way, he draws the 3 cards reward).
Based on this, it appears that any time a player's card or ability directly causes the elimination of another player, the player is responsible for the elimination (and draws the outlaw bounty if the eliminated player was an outlaw). Thus, both Backfire and Henry Block's ability would allow a player to draw three cards if that player eliminates an outlaw using them.