Can't find a reference to this in the FIFA rules, but here's what the USSF's Advice to refs has to say about this:
11.11 DEFENDER LEGALLY OFF THE FIELD OF PLAY A defender who leaves the field during the course of play and does not immediately return must still be considered in determining where the second to last defender is for the purpose of judging which attackers are in an offside position. Such a defender is considered to be on the touchline or goal line closest to his off-field position. A defender who leaves the field with the referee's permission (and who thus requires the referee's permission to return) is not included in determining offside position.
According to USSF rules, offside was the wrong call. Not sure what UEFA or FIFA has to say. Oh, and

for Liverpool grabbing an away goal.
Did not see the play but the advice to referees for USSF is the same as FIFA and UEFA rules. A defender stepping off the field of play must still be considered when determining the second to last defender. As well he could be cautioned for unsporting behavior.
The difference about the play today is that Carragher did not step off the field intentionally. He was carried off the field by his momentum as he was trying to make a play on the ball.
as a striker growing up, i was always taught that if you find yourself near the endline and the play is coming back to you, step off the field to avoid being in the play for offside. Then once its cleared hustle to get back onside. Dont see why it wouldnt be the same for the defense. Although a player is not offside if he is not in the play, so i guess thats your difference?
Any player that deliberately steps off the field of play to influence the game in any way (e.g., to create or avoid an offside call) is committing an foul that is punishable by a booking. So in the example you give, you could be given a yellow for doing that. If a defender did that intentionally, the play would not be offside, the ref would let the play continue, and the defender who left the field would be booked after the play had run its course. As for yesterday, Carragher was off the pitch because of his pursuit of an Arsenal player, so he wasn't commiting a foul. However, for the purposes of offside, he is considered to be on the endline nearest to where he is located off the pitch. Therefore, Bendtner was not offside, and if he hadn't had been such a klutz, Arsenal would have scored and won the match. Oh, and Wenger is a whiny little #####. They win that game if they finish their chances. He should complain less to a microphone and more to his strikers.
Never once got booked for it and used it often, esp when the clear was poor and the ball came back quick.Scenario, make a run down the field, cross the ball, no one there. def starts to push out and you are stuck near the endline and the ball comes back, step off and let the play go until a good clear. then hustle back up field to get back onside. Never once called for a foul a booking, etc. It was taught and was a generally good practice. And I have seen it in the EPL.
Not saying i am right or that the refs are wrong or the rule is false or anything. Just pointing out that I have never seen it called that way in my years playing or watching....
Maybe the rules were different for my era... but I was taught to obviously take yourself out of the play when you're caught in the offsides position. Not run out of bounds, but just run back to being ON-sides, making no effort or attempt to move towards the ball or play. IIRC, you had to be a part of the play-either directly, or indirectly to be off-sides... no? I'd go as far as holding my hands up (ala paintball, when you've been "killed" already) when I ran back. Seems to me, that's what I see the pros do as well (not the upheld hand thing). Gotta say- never seen anybody go out of bounds the way you're describing.

Wish I had seen the Carrack play, so I know more what you're alking about.
When I was 18 or so, at some point in an attmept at goal, I ended up actually IN the goal. Instead of waiting for the play to finish, I stupidly tried to get immediately out and back onto the field. I had to duck to avoid getting hit by a team-mate's rocket of a shot off the rebound from my initial attempt. The goal got called back because I was offsides, having just crossed back over the goal-line into play.