In the Flat Earth model, the sun is much closer to earth than in the globe model and so it's more of a giant lamp and there's only daylight in it's light cone. Look at the YT channel Vibes of Cosmos for some examples of how sun and moon work according to Flat Earth: https://youtube.com/@VIBESOFCOSMOS (hope this is ok to post, but I think it's rather uncontroversial)

(I'm not an expert, just followed the rabbithole a few years back. Some people researching the topic are actually quite nice and raise some interesting questions. They're not all right wingers (though often Christian conservatives) and most of them are convinced that Flat Earth Society is a psyop.)