How does the app know a note has a newer version?

Two ways. (1) Whenever it can reach your Supernote and you have the preview open, it does a quick, cheap check of the file's size and date on the device (no download) and raises the cue if they've changed. (2) Even when the device is offline, if a to-do linked to that note was created or edited more recently than the app's saved copy, that's a strong sign there's newer handwriting waiting, so it flags the note as out-of-date until you next pull it. That offline flag survives restarting the app.