Does activating a Repeat on a card count as playing it twice?

No, activating Repeat on a card does not count as playing it twice.

According to Rule 746.3.a: "Regardless of the number of times a spell's instructions are executed with this keyword, the spell is only Played once."

When you pay the Repeat cost, you are paying an additional cost to execute the spell's instructions an additional time as part of the same chain item. Because the card is only played once, it only counts as one "played" event for the purposes of any game effects that trigger when a card is played.