What is the name of the process for creating a new DB from an exported snapshot?
If you are just looking for nomenclature to name the process, I would call it

DB Sandbox Provisioning
DB Self-Service Provisioning
DB Self-Service

If you need an actual process supplied by Amazon, you could look into

AWS Control Tower
AWS Service Catalog

If you are planning to script this yourself, there is always the AWS CLI.

create-db-snapshot
restore-db-instance-from-db-snapshot

Check out the Amazon Docs for further info.