[FEATURE] Use FAL API for moving file across storages

Change-Id: I09bd9142a0af5030549ca246e7f6214defa03681
Releases: 2.0.0
Resolves: #54660
Reviewed-on: https://review.typo3.org/26588
Reviewed-by: Fabien Udriot
Tested-by: Fabien Udriot