Suppose you have one external HDD which you connect once a week to your laptop and backup all your important data (e.g. photos/videos) to this external HDD. Now suppose that you have already some backup on external HDD but in the mean time you have reorganized your multimedia collection on laptop (added/deleted/renamed/moved files/folders etc.). I'm looking for software which is capable to "understand" those changes and backup only differences.
There is no need to copy again files which were just renamed or moved to another directory (simple move/rename should be sufficient also on external HDD because those files already exists here), only the new files should be copied.
Some statistics would be also fine, e.g. before I plug external HDD I would like to see which files will be copied which were deleted, moved etc.
I'm looking for free or open source solution. Multi platform software (Windows/Linux) would be fine but not necessary (I'm not afraid of doing some hacks like running in cygwin or under Ubuntu terminal in Windows 10 if this is only option).
History of backups would be fine but not necessary (e.g. show me which files were inside this folder at given date)
As a result of this I expect that when only small changes were done in data then backup process should be really fast.