If you somehow control two copies of Wayta, Trainer Prodigy, a creature you control being dealt damage causes abilities to trigger three times, not four. A third Wayta, Trainer Prodigy causes abilities to trigger four times, a fourth causes abilities to trigger five times, and so on.

Wayta, Trainer Prodigy's last ability doesn't copy the triggered ability; it just causes the ability to trigger twice. Any choices made as you put the ability onto the stack, such as modes and targets, are made separately for each instance of the ability. Any choices made on resolution, such as whether to put counters on a permanent, are also made individually.

Wayta, Trainer Prodigy's last ability affects only triggered abilities whose trigger conditions refer specifically to damage being dealt, such as the ability granted by Mephidross Vampire or the last ability of Wrathful Raptors. It does not affect triggered abilities that would trigger because of the results of that damage. For example, if you control an Ajani's Pridemate ("Whenever you gain life, put a +1/+1 counter on Ajani's Pridemate") and you activate Wayta, Trainer Prodigy's second ability targeting a creature you control with lifelink and another creature you control, the triggered ability of Ajani's Pridemate will still trigger only once.