Hello,
These options work as follows:
System "Auto-Sync" setting - whenever your system accounts (Android settings > Account page) are synchronized by Android, an event is triggered to synchronize the email accounts you have in Aqua Mail.
The app resuming - whenever you open Aqua Mail to the foreground
Device reboot - pretty self-explanatory
If you are having issues with these events, it is worthwhile to disable battery optimization for Aqua Mail. It could be just that your battery optimizer is killing the app's event monitoring.
There are also other ways to save battery:
1. Using push mail (if supported by all of your accounts) and limiting scheduled sync to once or twice a day.
2. Limiting how much is downloaded every scheduled sync event (Settings > Network and Settings > Manage accounts > your email > Options > Preload)
3. Limiting "messages to sync" to a lower number/value in days.
4. Turning off Push Mail and only using scheduled sync (this means you don't get notifications instantly when new mail arrives)
5. Using Nights and weekends (Aqua Mail Pro only) to turn off sync/push in specific intervals, like during the night.
6. Setting up per-account rules for sync/push, in case you only need one or several accounts to receive mail often/immediately but not all of them.