The second of those, however Ven is a different case because he didn't "lose" his heart the normal way. Ven's existence "stopped", or rather fell comatose because his heart was critically injured, not lost to the darkness.
If one pays close attention in the scenes of Aqua's ending and the beginning of Final Episode one can even see that Ven still has his heart all the way up until Aqua leaves him in the chamber of waking, as he can still summon his keyblade to open a door to Land of Departure.
The reason Ven went comatose is not because of his heart physically leaving the body, but because it was so heavily injured and weakened it couldn't maintain an awake state anymore.
So Ven's heart is "lost", but not in the sense of losing a heart to create a heartless.
When Aqua left Ven alone in his chamber, his heart couldn't struggle against the wound any longer and started to fade (this is not the same as creating a heartless, losing your heart doesn't always mean the creation of a heartless. Riku's heart got kicked out of his body in KH 1 too and it also became no heartless!).
The extreme pain and sadness Ven felt at that time reached Sora through their already established connection, and his voice guided Ven's heart to Sora's, where Ven's plea to stay with him was accepted by Sora.
So Ven's heart went to "rest", so in essence Ven is really "sleeping".
So, considering Ven's case, the second term should probably renamed to "Existence Sleep!", lol