Does the “real time” timer include all of the time you spend between stages? If it doesn’t, why do we need to include both? Also, which is shown as public, in-game time of IRL time? There is like 5 minutes (like 400 speedrun spots) of difference for mine, so I am a bit confused and it’s kind of important.

You are supposed to leave Time blank and only fill in In-Game Time if you read the game rules. So yeah don't worry about real life time just look at the final in-game time when you finish your run.

Just want to confirm, yes Celeste doesn't use real time as a timing method at all and all submissions should leave real time blank. Just enter what you see for the chapter/file time (depending on whether it's an IL or full game submission). Each category's rules will spell out exactly what you need to submit