is a specific kind of: amplify, exaggerate, hyperbolise, hyperbolize, magnify, overdraw, overstate.

is a broader term that covers: puff | crow, gloat, triumph.

derivation: bluster, bravado | boast, boasting, jactitation, self-praise | brag, bragging, crow, crowing, gasconade, line-shooting, vaporing | bluster, braggadocio, rhodomontade, rodomontade | vaunt | blowhard, boaster, braggart, bragger, line-shooter, vaunter | blusterer, loudmouth.