I always got the impression that the “next week on” spots, unless deliberately worked into the show as a joke, were done by the network and aren’t part of the episode, whereas recaps are nearly always part of the show.
As for how they should be on the DVD: leave them in, but give them and the opening title sequence their own tracks. It’s classy and professional that way.
Previously, on Third Watch [five minutes of stuff from the last few episodes which may-or-may-not relate to tonight’s episode]
Tonight, on Third Watch, (two minutes of spoilers for the episode you’re planning to watch right now], obliging you to look away, put your fingers in your ears, and go “LALALALALALALA!” until you detect a little Crystal Method leaking through and can face forward without wrecking everything
I could live without that, yeah.
I’m almost certain the DVDs for The Sopranos include them. Like Gyrate said with Babylon 5, they’re separate menu items, but they are there.
IIRC there was (at least) one episode of Lost where the “Previously on” bit included something that wasn’t actually previously on. It had been filmed then cut for time but needed to set up something in the current episode. Can’t remember which one it was though.
Lost, in particular, is more likely than most shows to reference something from many episodes back. In those cases, the “previously” bit serves as a nice reminder.
Like the final episode of The X-Files, which, if memory serves, consisted of 90 minutes of “previously on” and 30 minutes of bitter disappointment.
Weird. I was just thinking the other day (while watching some Supernatural reruns on the DVR) how great the “road so far” segments on the show are and how this show in particular just wouldn’t quite be the same without it.
I say shows should include these on DVD, but adding a chapter break so you can skip them if you want is a painlessly easy thing to do and should be the norm. Likewise, there should always be one right after the opening credits. Why they don’t always do this is beyond me.
Most shows, they can reference things from any previous episode, not just the immediately previous one, and seeing what’s significant for the “previously on…” helps set you up for the episode. Battlestar Galactica also had a nice touch in that the “previously on Battlestar Galactica” line was spoken by whatever character was most significant in the episode, so you could say “OK, this is going to be a Baltar episode”, or whatever.
I’d love to be the show-runner for some long-lived show and make it a running gag to show “previouslies” that had nothing to do with the upcoming episode.
Doctor Who episode about Rose and her father? “Previously …” Daleks from 1966.
Star Trek episode about Tuvok going through ponn farr? “Preiously …” tribbles.
Jack Bauer getting ready to put on his ninja-Batman armor? “Previously …” tea towel in the back seat of the limousine.
Well, maybe that last one isn’t entirely unrelated. 
And it is like they know it, which is why they include the option to get rid of it. I also like them, by the way. That is a show I’d keep them on for.
I’ll show you one if you have not seem them.
It bugs me that my DVD copy of the Buffy episode “The Gift” (the 5th season finale) lacks the ‘Previously on’ because:
A. It gave a summation of the entire series up to that point
and
B. It builds up to the opening scene of the episode. Without it, the opening of the episode seems strange.