Unfortuntately, this game doesn't load, so I can't check the specific interaction. But generally what you describeded as happening is correct.
Dominion cards are carefully worded such that if a card is trashed that belongs to the deck of a player, it is explicitely that player that formally does the trashing. For example, Pirate Ship reads
...each other player reveals the top 2 cards of their deck, trashes one of those Treasures that you choose,...
So, when playing Pirate Ship, while you
are the player choosing
which Treasure to trash (if there were 2 revealed), it is the other player
that does the trashing
Since Priest cares about who trashes the card:
... For the rest of this turn, when you trash a card, +$2.
your Pirate Ship atacking doesn't trigger your Priest's effect.