[TASK] Drop usages of TYPO3_MODE and TYPO3_REQUESTTYPE

Drop all usages of TYPO3_MODE and TYPO3_REQUESTTYPE by
using the ApplicationType helper class when frontend
or backend is detected, and by directly using the
applicationType attribute of the request object in a
couple of special cases that check for backend ajax
or install tool.

Resolves: #92953
Related: #92951
Related: #92947
Releases: master
Change-Id: I98c9d5ef0e7a6409b01188ddd0bbcf94f159cbcd
Reviewed-on: https://review.typo3.org/c/Packages/TYPO3.CMS/+/66895


Tested-by: TYPO3com <noreply@typo3.com>
Tested-by: Simon Gilli <typo3@gilbertsoft.org>
Tested-by: Oliver Bartsch <bo@cedev.de>
Tested-by: Benni Mack <benni@typo3.org>
Reviewed-by: Simon Gilli <typo3@gilbertsoft.org>
Reviewed-by: Oliver Bartsch <bo@cedev.de>
Reviewed-by: Benni Mack <benni@typo3.org>