EBS volumes are like external harddrives. You can attach and mount them on the system, but if you want the data to be "persistent", you will need to mount bind the volume and configure it in such a way that the database and all files moodle uses will be stored on the volume. It's a little tricky but possible.

The easier alternative would be to use TKLBAM to backup the system. Once all data is backed up, you can terminate the instance. Later you can "restore the backup to a new cloud server", which will essentially launch a new moodle instance, configure TKLBAM and perform the restore automatically on boot.

BTW, persistent instances (i.e. EBS backed instances) are currently under development, and will provide the ability to stop/start the instance which is what you are looking for. But keep in mind that even though the instance will be persistent, EBS volumes do sometimes fail, so it's important to have backups - and TKLBAM makes that easy.