Kevin Donahue has a very good explanation here
I like his color coding of the TiVo logo plus he's got some real thoughts on the differences between TiVo and other DVR/PVR's. That's the big issue I have as well. People have a DVR and they say they've got a TiVo. I guess it's like calling tissue paper a Kleenex but really different in the case of TiVo since the functionality is waaay different.
Excerpt from Kevin's post:
What's the difference between a DirecTV TiVo and a DishPVR?
There are some core differences which people don't seem to understand. Most DishPVR user think TiVo is a commodity term to describe a PVR, like "TV", "DVD", "VCR", etc. If you ask them if they have a TiVo, they'll say "yes" but they actually have a DVR. If you're looking at the two (DishPVR vs. standalone TiVo or DirectDVR/DTivo), it's important to understand that each one is a Digital Video Recorder. Only the TiVo units are TiVos. Why the big deal?
Both TiVo units and DishPVR can perform the "TRICK PLAY" features I described above. Both types of units can record. That's where the similarities end.
DTivo/DirectDVR can record two signals at once - DishPVR cannot. Even better, your DTivo/DirectDVR lets you watch one saved program and record two others at the same time. This is a key feature in our house when West Wing goes up against Amazing Race or when Monday Night Football is on opposite some sappy, girly movie. Another drawback (as I understand it) is that you pretty much have to watch what you're recording. You can't record Friends and watch something else. If you want to record Friends, you have to sit there and watch Friends. Boo!
Also, DishPVR units do not have the TiVo service. That means they cannot search out programming, accept thumbs up/down, wishlists and other features described above.
Essentially, the DishPVR is a fancy VCR - nothing more. You still need to figure out what you're going to record and when you're going to record it.
One "feature" of the DTivo/DirectDVR is that you could upgrade/add hard drives if you were so inclined. I say that carefully, because you could void your warranty. But, if you have to have a larger hard drive, its definately possible.
Even if you get a DishPVR for free, it really doesn't do that much for your TV experience other than let you fast forward through commercials. The smart money is on the TiVo system which offers a lot of service features that truly change the way you experience television.