After hours of experimentation I've figured it out. I've tested it again and again and it works just like the manual says, in single player.
There are garages all over town that each character can buy, garages that are not tied to safe houses. You just have to put your customized vehicle in one of those purchased garages immediately after you customize it and not in a safe house garage. If you then take that customized vehicle out of the purchased garage and abandon it somewhere it will be in the impound the next time you load the game and it will retain your customizations. I'm not sure how long it takes for it to go to the impound if you don't exit and restart the game, though I've read of other players having to wait several in-game days.
So in essence, the impound is your customized stolen-vehicle garage, so long as you have put that vehicle in one of those non-safehouse garages before taking it out and abandoning or tearing it up. You just have to exit and reload the game for it to show up there. Gone are the days of causing mass mayham and then losing all that progress / kill count / stars evaded because you had to reload from a save to get your aweseome vehicle back.
The impound can only hold 2 vehicles so I'm guessing that, as some other players have said, if you already have 2 vehicles in the impound and you abandon/destroy your customized vehicle you lose the customized vehicle, though I've yet to test that part.
Another thing I've yet to test is destroying the vehicle with a grenade, though I'd bet it would work just the same since I can do mass amounts of damage to the vehicle and it still goes back to the impound. It doesn't work if you sink the vehicle in water though, at least not in my experiment.
Still up for experimentation:
- If you have 2 vehicles in the impound and you abandon a customized vehicle, does the customized vehicle, being owned, take the spot of one of those 2 that are already in the impound (and probably not owned) or do those original 2 take priority. (And the related question that seems to be more a bug than something that can be experimented on, why do stolen non-owned vehicles show up in the impound?)
- How long does it take for the customized vehicle to show up in the impound if you don't exit and reload the game?
- Further experimentation on sinking a vehicle in water is needed as I only tested it once after buying the garage and that was so many 'experiments' ago that I don't remember if I already had 2 vehicles in the impound when I did it.