Originally Posted by KneeKnee
No, you do the speed run as one playthrough after you do the no deaths + 6 chaos emerald playthrough. You should be saving often during the no deaths run, so you just make sure to have 50 rings at the end of the appropriate levels and save as soon as you're in the bonus stage. If you fail at getting the emerald, or you die at all, you reload. This saves you a playthrough.
You want to do the speedrun without getting the emeralds because those stages count towards your total time.
I mean 3 playthroughs: 1 without dying, 1 doing all the collectibles and 1 for speedrun.