[TASK] Remove dummy files from ext:tstemplate_* extensions

* Remove dummy files from ext:tstemplate_* extensions
* Remove old class names to the new ones
* Adjust function names for the backend module
* Adjust AJAX save type

Change-Id: I560c1c0e7bceb271eaba1a1d660cec71614df64a
Resolves: #48157
Related: #47756
Releases: 6.2
Reviewed-on: https://review.typo3.org/20766
Reviewed-by: Anja Leichsenring
Tested-by: Anja Leichsenring
Reviewed-by: Christian Kuhn
Tested-by: Christian Kuhn