No, that is not correct. A human player is not eliminated from the game until he has lost his last remaining province, either core or non-core. And even then, any units the player may have remain on the board until the player is inactive for two days. Those orphan units may continue to fight after their country's last province has been lost, but that is usually an exercise in futility, but on rare occasions some players have staged a comeback when they have large numbers of remaining units.
Please note that a somewhat different set of game dynamics apply to AI countries. When an AI country loses its last province it is immediately eliminated from the game, and all of its remaining units are immediately eliminated and removed from the board. AI countries do not have the ability to attempt a comeback once they have lost their last province. This was the result of changes implemented last year; previously orphan units from all eliminated countries remained on the board until they were destroyed.
The most important aspect of core provinces is that they produce resources -- food, goods, metals, oil, rare materials and money -- at four times the rate of captured (i.e. non-core) provinces.
