What’s meant by “the people of Lot”? The Bible names them as the Moabites and Ammonites.
The Sodomites were not Lot’s people; he lived there before it and Amorah (Gomorrah) were destroyed.
From what I gather from the very brief research done, I take it to mean Lot’s neighbors, or people that lived in the same town as Lot, not necessarily his family.
Perhaps a Koranic expert on this forum can shed some light on your question.