I'm not exactly sure about this, but maybe the ability to activate PoD after a Special Summon depends on how exactly how the Special Summon was negated. (Effect or Activation negation)
For example, you control a Lumina, you use its effect to revive a Lightsworn monster from your graveyard. Your opponent uses Effect Veiler to negate its EFFECT, then you will not be able to activate PoD later.
However, if your Lumina's effect had its ACTIVATION negated by something like Solemn Warning or Herald of Orange Light, you should be able to activate your PoD later since technically you didn't attempt to Special Summon this turn.