I am looking for a backup tool that synchronizes folders, such that changes to one folder are applied to the other one, and vice versa.

Let's say I have a folder containing files a and b and sync it to an USB stick. Then the USB stick also has files a and b.

If I create a file c in the original folder, and a file d on the USB stick, the tool should see the two files and copy c to the stick and d from the stick, such that there are four files in both folders.

But if I delete a file in one folder, it should delete the file in the other folder. I.e. it should not compare the two folders directly, but compare each folder to how it was before the last syncing.

If I rename the file a to x in the original and sync again, the tool would see that the file was renamed (because x has the same content/hash that a used to have), so it renames a to x on the USB stick. It should not delete a and copy the new x there, since copying takes a lot of time with large files. With directories, too. If all files are in one subdirectory and I rename it, syncing should rename the corresponding subfolder without copying everything.

It would be like git, but without storing the entire history of the file content, so that when the file is changed or deleted in all repos, the old content is gone forever.

The rsync utility is pretty darned good. This technology is the basis of many venerable backup systems, including Apple's Time Machine.

I do not think that detects renames. It just copies the file again

And I always get confused by the trailing slash and then I end up having every file twice.