This makes the most sense. Start a timer when the UI actually hands control to the player whose move it is, stop the timer when they've completed their move, an

This makes the most sense. Start a timer when the UI actually hands control to the player whose move it is, stop the timer when they've completed their move, and simply subtract that from their remaining time. The interval gets sent to the server and relayed to the other player to update their opponent's clock accurately.