Can I change the order of the test types in the ribbon?

Yes, you can.  The procedure is explained in the instructional video on Test Management.

This is how it works. Click on this arrow to get a list of all test types:

You can pick a test and change its place using the up-down arrows.

The test types will appear in this order on the ribbon. You can also choose to give priority to the test types which contain results (as marked by the green check in the ribbon).  For that purpose, just check this box:

Now test types without results will be moved backward in the ribbon. You can also choose to give even more priority to test types containing results of today. For that purpose, use the other checkbox.

Now the ribbon shows the test results of today (orange) first, then the other test types containing results (green), and the empty test types at the end.

This feature is not available in the web viewer.