Don't run extension cords under rugs — period.

Neither is safe. The question assumes one is better than the other, but you’re asking about a fire hazard, not a shopping decision.

Flat cords are slightly less likely to create a visible bump, but they still trap heat. Carpets and rugs act as insulation. Any extension cord — flat or round — can overheat under a rug, especially if it’s carrying a decent load. Round cords are usually thicker and worse about this, but flat ones aren’t safe either. The insulation can get damaged from foot traffic over time, and you won’t know until it arcs or shorts