FWIW "playing in the Sun" kinda sounds like you're somehow managing to exist inside the sun's mass and playing there, whereas "playing in the sun" clearly refer

FWIW "playing in the Sun" kinda sounds like you're somehow managing to exist inside the sun's mass and playing there, whereas "playing in the sun" clearly refers to merely being exposed to sunshine and "playing in The Sun" sounds like you're trying to say that The Sun published an article about your play.