I personally think of TestSuites as a way to group a bunch of tests into a common test environment rather than as a way to reuse a TestCase for testing multiple parameters. But I can definitely see the value of being able to set the displayed name of the test case on the VI Tester GUI and how it would help handle the situation you raise of re-running the same TestCase with multiple parameters.
The simplest way that I see to solve this problem is to just have 3 TestCases that share some common test code so that you can have easy to debug and use the three test cases to define the tests for each logger.
In this scenario, I think you'd have a test folder hierarchy like this:
Tests
Tests\Logger\DiskLogger1
Tests\Logger\DiskLogger2
Tests\Logger\InMemory
Tests\Logger\Common
The simplicity of this is that you can specify the specific tests for each logger, you don't have to worry about it being too complicated and you can add all three TestCases to a TestSuite if desired so that they can share some resources or be grouped for logical organizational reasons (or you can skip the TestSuite if you like). Most likely, this is the way that I would approach that problem.
The other way that is possible to approach the problem is to use test case inheritance. VI Tester does support TestCase child classes but you will have to make some tweaks to your code to make this work.
On disk you'll have something like this:
Tests\Logger\DiskLogger
Tests\Logger\DiskLogger.1
Tests\Logger\DiskLogger.2
Tests\Logger\DiskLogger.InMemory
where DiskLogger.1, DiskLogger.2 and DiskLogger.InMemory are classes that inherit from DiskLogger (which is the TestCase with all of your test methods). You can create the child classes just by creating classes from your project - you don't need to create them using VI Tester. You'll need to modify the test methods to use dynamic dispatch inputs and outputs (which will also require that the re-entrancy type changes to 'shared' - LabVIEW will warn you about this and its pretty easy to update). You'll also need to create over-ride VIs for any tests in your child classes that you want to execute (currently by default, VI Tester is conservative and assumes that if you don't specify a test VI to over-ride then you don't want to execute it from your child class). When you create a "New VI for Override" and specify your test method, by default the implementation is to call parent method so this can be done pretty quickly or via scripting.
After all that work, you can use your original TestSuite (the first image) and wire in a bunch of your child TestCases and each child will show its class name on the VI Tester Tree.