A Cuban cigar is made from tobacco grown on Cuban farms and rolled in Cuban factories. That is the only definition that matters. Cuban seed tobacco grown in Estelí does not make a Cuban cigar. A roller in Miami who used leftover tobacco bought before the embargo in 1963 was making an American cigar with Cuban tobacco; that is a historical detail, not a category that exists today.
Cuba nationalized its cigar industry after 1959. The United States embargo then removed Cuban cigars from the legal U.S. market. For the rest of the story, see Cuban vs New World and the cigar.
This page does not rank Cuban cigars against Dominican or Nicaraguan ones. They are different industries with a shared history.