Is a player eliminated when he runs our of centers or when he runs out of units?

My inclination would be to say it's when he runs out of centers, meaning the players in your example were rightly given defeats rather than survivals.

But you say the rule is that players are defeated upon losing all units, rather than losing all centers? Can you provide your source? I think I lost my hard copy of the rules. If you are correct, then it seems there is a loophole the site adjudicator does not currently observe.

The rule seems to be observed properly now. Here is an example of a game in which Turkey loses both of his last two centers in the final moves of the game. In this case, his units actually do not have anywhere to retreat to and hence will disband, but the retreat phase has not gone through yet and hence he is given a survival with no centers.

http://webdiplomacy.net/board.php?gameID=145452#gamePanel

Your second example is definitely adjudicated incorrectly, though.

We know that victory cannot occur until after the retreat phase, since situations arise where a dislodged unit may retreat into an empty center to deny a player the victory. There retreat phase must happen before Russia is given the victory, even though Turkey has no valid retreat options. In this case, Turkey should receive a defeat regardless of whether we are going by centers or units. Russia wins the game AFTER the retreat phase, by which time Turkey has neither centers nor units.

It seems to be being observed properly by the end of 2008.

And now I'm thinking that ownership of centers should not be determined until after retreats, which is certainly not the case in your examples

Yes, but I am thinking that the final move seems to be different. You make a great point, but the in the final move of the game, the rule seems to be that as long as there is no way for a dislodged player to retreat into a center occupied by the winning player, which would bring them below eighteen supply centers, then the game is over and the retreat phase does not have to take place. That way, the game can end without having to adjudicate the retreats.

I understand that you could argue that you would have to adjudicate the retreats to know that that is the case, but the rule seems to be simply that as long as it is not the case, then the game ends prior to the retreat phase, not after. I say this because on my JDip program, it adjudicates the end of that game, August Rumble, exactly how Webdiplomacy does. Russia wins prior to the retreat phase, not after. The game ends on retreats and Turkey survives the game.

In early 2008, Webdiplomacy seems not to have been observing the rule this way, as I talked about above, but this is how it observes the rule now. I think that it was changed at some point in 2008 because I found a game that concluded in November of 2008 in which the rule was adjudicated in this manner.

IIRC, a bug still exists that allows for a game to end after the fall turn but before retreats. A current admin can correct me if I'm wrong.

Interesting discussion. I was always under the impression that players 'surviving' without SC's if eliminated in the last year was an unintentional quirk, presumably due to webDip engine restrictions. I always assumed that such players were (properly speaking) eliminated per the Diplomacy rulebook.

"We know that victory cannot occur until after the retreat phase, since situations arise where a dislodged unit may retreat into an empty center to deny a player the victory."
This is a separate issue. As 2ndWhiteLine says, webDip incorrectly ends games before the Retreats/Disbands phase.