It really boils to: From an organizational perspective, what is the database?

Is it just the logical model, objects, and data; or is it the hardware, network, SQL, and ETL processes too?

Strictly speaking, I think the DBA should be someone who manages the objects and data within one or more databases, but not necessarily the system administrator of the server instance or operating system. Also, one can be the administrator for database [Sales] but not the database [HumanResources], even if both are contained in the same server instance. For example, administrating the PeopleSoft, SharePoint, and MS Exchange databases can be a specialized full-time job by itself, while the proprietary databases built in-house that support the organization's line of business are something else entirely. In the SQL Server, this would be a member of the DBO role, perhaps with some additional granted permissions so they can manage jobs, view execution plans, etc.

Ideally, the system administrator (members of SYSADMIN role), should be confined to a much smaller group of people who manage the hardware, operating system, and networking environment. In a large organziation, their role is at the enterprise level, and they should not necessarily be familiar with the data model and processes any database. Their job is to make sure the databases are available, secure, and backed up.

If someone is expected to wear both hats, then their job description should include both Database and System Administrator.