You could argue that everyone on earth is just postponing death.
I guess that statement implies that the person is ready to die, and it was the family that was fighting to keep her from rest. I imagine a scenario where a patient has signed a DNR but somehow gets put on a vent etc against their wishes... *that* is postponing a death.
If the person is fighting though, then yes, they are fighting to live. This really seems like a glass is half empty, glass is half full discussion. It just depends on how much tenacity you have for life.