Is it limited to Time Attack only, or can we also do the Story Mode version?

if you wanna do it outside of time attack ILs there's an act select and full game run IL leaderboard in https://www.speedrun.com/s3airce

All Individual Levels on the main leaderboards must be done on Time Attack mode, that includes Sky Sanctuary. However, if you want to run as Sonic & Tails or modded characters, go to the category extensions leaderboards.