SQL Server Agent Server Login Account
The SQL Agent uses whatever is assigned as the login on the Windows Service.
To access Windows services and see, click the start menu and type 'Services'.  

Locate the SQL Agent and view the properties.
If the account is the Local System, then it is actually using the credentials of the machine.  You confirmed that already when you saw that the DOMAIN\SERVERNAME$ was returned.

(That is the account for the machine that it's hosted on.  All PC's have a local system account and it gets added when SQL is installed.  I believe the system account inherits sysadmin privileges which is why you probably don't want to leave it as the default. NT AUTHORITY\SYSTEM I believe is how it gets added to the logins in SSMS.)

If you want to assign different permissions which is a good idea, you can use a different login, usually a service account or master service account. 

Alternatively create a credential in SQL server assigned to an account that has permissions.  Then create a proxy in the SQL Agent and you can use that to run the job. 

According to MS best practices is to always assign the login credentials for the service accounts using SQL Configuration Manager.  While I've never run into a problem doing it other ways, if MS says so, it's probably correct.