K-Lo: The way it has always been explained to me is that the player "Buying the Button" is paying the BB for the player that would otherwise be responsible for it. In my example which you quoted below, the player in seat 3 is the BB and busts out, leaving the table. Without the Buy the Button rule, normally that seat would remain vacant for two hands, until the Button passes to seat 4. If a new player shows up for that seat he has to wait.
With the Buy the Button rule, that new player can post both blinds, the BB is live and the SB is dead, get a hand and on the next hand get the button. In effect, the new player has posted the BB for the player in seat 4. On the next hand, the player in seat 3 gets the button, seat 4 is the SB and the BB is in seat 5.
So yes, the player in seat 4 gets a free BB round. With this rule in effect, there is no "dead button" as would normally be the case.
Anyhow, this is how it has always been explained to me.
Chet