SQL Server: Clarity on which service user does what (backups, etc) RE accessing network shares
The Service Account for SQL Server needs to be granted access to the network share in question.  

This indicates a need for you to configure SQL Server to start with a Windows Active Directory account.  If you don't have access to a domain account, you will need to create an account on the SQL Server machine with a matching account name and password on the target network machine.

Use SQL Server Configuration Manager to modify the startup config of the SQL Server Service.



The only way to allow a service on one machine to access a network share is to provide network access with a username and password.  Make sure the SQL Server service can access the network share before restarting SQL Server by logging onto the SQL Server machine using the same username and password, and confirm you can see the network folder.

You also need to ensure you aren't trying to restore from a "mapped" network drive, since the "mapping" won't necessarily be there for the SQL Server service to "see".  Instead of using a mapped drive letter for the restore, use a UNC name, such as \\server\share\folder\backup.bak