Curt wore the I-C belt the same way Flair wore the NWA title in regards to the prestige they brought to the belts. Both men looked like champions, and looked like a gold belt should be worn around their waists.
Savage was similar.
I never cared for guys who carried the belt to the ring with them. I know that, for some guys like Austin, it was in accordance with the character, but it still never sat right with me - dragging a championship belt on the ground behind you.
How many of you remember that the original "Attitude Era" heavyweight title strap was blue? It didn't last long, though. Austin beat the hell out of that belt dragging it, throwing it, spilling beer on it. It soon stained to the point the leather appeared black. After changing the WWF logo from the block to scratch lettering, they modified the buckles and switched to a black strap.