Option 1
Create two different Test Cases, one for SQLServer and one for Oracle and then to automate first by Test1 and the second by Test2. So you can run both of them by a single build definition which is using DefaultLabTemplate11.
Using suggested approach the test code will not have to implement additional logic in order to recognize which database they should run on.
You said that you want to use the Test Case for both manual and automated testing, so I suppose the Test Case you have today consists of several Test Steps describing how to run tests manually.
In this case you can clone existing Test Case and use shared Test Steps but you will still need to update both Test Cases when you add/remove some Test Steps... this will be the only disadvantage of suggested approach.
In order to get rid of this disadvantage you could create "Manual database tests" Test Case, assign two Configurations you already have to it and use it for manual tests only. This Test Case will contains all Test Steps for the tester.
Then create two Test Cases I've described above, automate them and use them for automated tests only.
Option 2
If your test environment is a virtual machine you could avoid creating two Test Cases:
- Store the connection string in a file on your test environment and let the test read this file.
- Create a two snapshots: the first with the file containing SQLServer connection string, the second containing Oracle connection string.
- Create two build definitions one reverting the environment to the first snapshot and the second reverting it to the second snapshot.
This way you can keep your single Test Case but on the other hand having two build definitions can become unhandy if the build definition is supposed to build the source code before deploying and running tests.
Well... the third option will be to implement additional logic in the test in order to recognize which database it runs on.
But in this case you will also have to create two build definitions since you have two Configurations and you can only select one per build definition.
(I hope I've managed to describe it well, don't hesitate to ask it something is still not clear).