[BUGFIX] Streamline element browser templates

The templates used for the element browser are now separated into
several templates, based on their usage.

The element browser used in combination with FAL does not share the
template with the folder-based element browser anymore.

Resolves: #97933
Releases: main
Change-Id: Ia1c4278c8fb656c8979dec9b60f2f892af18bd4d
Reviewed-on: https://review.typo3.org/c/Packages/TYPO3.CMS/+/75172


Tested-by: Benni Mack <benni@typo3.org>
Tested-by: core-ci <typo3@b13.com>
Tested-by: Georg Ringer <georg.ringer@gmail.com>
Reviewed-by: Benni Mack <benni@typo3.org>
Reviewed-by: Georg Ringer <georg.ringer@gmail.com>