What happens when a card with undying dies that has a +1/+1 counter put on it by something other that undying?
Undying doesn't care how the +1/+1 counter got there. Undying won't trigger.

Conversely, if Undying previously placed a +1/+1 counter on a permanent, but the counter got removed somehow before the permanent died a second time, Undying will trigger and bring the card back a second time.