Syncthing is very flexible in this point. I am using it to distribute, synchronize and backup my files between various devices I got (2 Desktop PC, Notebook and Smartphone). My personal files are synchronized between all devices, bigger archives like my RAW-files are synchronized between the two desktop PC.
For a simple backup solution you could use a cheap Raspberry Pi with an external hard drive connected as a backup storage with syncthing. Much more flexible than a NAS.
I do use syncthing a lot but it is not a backup. At least I can’t see it as such.
I would advise against syncthing as a backup solution. Without past snapshots, it won’t prevent you from losing all data once you sync deletions.
Borg-Backup or appropriately setup rsync, or arq, or restic are better solutions for backup.
My borg has continuous monthly snapshots from ever since I started using it, many years ago. Despite changing operating systems and file systems and computers and everything several times in the meantime. I can highly recommend it.
(syncthing is great for file synchronization, though)
It sounds like synchronization is the better bet. I just want to be sure that a file i just edited, is stored on an external drive as well as my computer.
it depends on what you want. When synchronizing, if you accidentally delete a file, then the sync happens, the file is deleted in both places. Sync is not a backup.
These two points are worth reading a few times. Syncing is one thing, backing up is another.
I’m personally a fan these days of restic but i know some others love borg.
I’ve got daily snapshots going back weeks that only take up as much space as files I’ve changed. You could set it to run on a schedule every hour (or less). It’s fast at finding differences and only needs space to store the diffs.
Good point. I really want copies of the files that I edit. Can’t afford to lose a synchd file if I delete a working copy by mistake.
I use the btrfs filesystem and its incremental send/receive capability.
my solution is
# The last / in source directory means don't create A6300 directory again under Destination A6300 directory
/usr/local/bin/rsync -avE --exclude=.DS_Store "/Volumes/HackSSD/A6300/" "/Volumes/DevToshiba/A6300"
/usr/local/bin/rsync -avE --exclude=.DS_Store "/Volumes/Data/A6300/" "/Volumes/DevToshiba/Data/A6300"
# in this case below NO / after darktable means create if not present darktable directory under DarkTabe Backup directory & copy
/usr/local/bin/rsync -avE --exclude=.DS_Store "/Users/raj/.config/darktable" "/Volumes/DevToshiba/DarkTable Backup"
This is part of a script which is autorun when I plug my USB backup drive and it will not delete any files in the backup drive only backup modified and new files. I am on OSX Big Sur
I’m also a great fan of borg. I make an automatic daily snapshot of my data and store it encrypted “in the cloud” (my own server). No problems for serveral year.
In addition, I sync the data from time to time to a second computer in my home with rsync. Just to be sure 
.
Put another thing up:
For years and years I use rsnapshot.
It is actually a script using rsync in the background.
Files can be read directly, only native Linux tools are used.
Is the original question related to a specific operating system?
I personally like multiple backup methods:
On Windows I am using FreeFileSync, Hardlinkbackup, AutoVer and Syncovery.
I use autover with my xmp files…that way if I screw up I can go back in time so easily to any version of my edit…
I should give this a try. I use rsync to external USB but I also do combine it with periodic duplicati for .xmp and a cloud version.
I use rsnapshot as well, but the original question was:
As far as I know, none of the tools mentioned recognise when a change to a file is made. That could be done (al least Linux) with some kind of filesystem watcher/trigger, can’t it? I’m not aware of any backup program doing it that way though.
I am not sure I am grasping the point here. Even if the FS would detect file change - it would need an action to make the snapshot (it does not make snapshot by itself so a user can somewhat step back every single transaction).
Since the user has to initiate the action (based on schedule or manually) what is the difference between FS approach and backup approach?
So yes - IMO backup does provide the functionality as described in the original question. It is rather - what tool works most elegantly for the user (and each has pros and cons).
I took it to mean “the filesystem notices a change, and makes a backup of the file” (so it would be at a filesystem level, and work in all programs). But maybe I’m out cycling 
No backup program I know does this. The closet would be a filesync program with a versioned trash. If properly configured (i.e. not default config) Syncthing can do that, maybe with some caveats.
I agree with everybody saying filesync is not a backup in the absolute sense, but can be used as a temporary backup or an intermediate step.
Basically that’s named “incremental backup” or “differential backup”.
See Incremental backup - Wikipedia and Differential backup - Wikipedia
Backup software I’ve used has generally had 3 different modes:
Backup: All new and changed files from source are copied over to destination. Deleted files on source do not get deleted on destination.
Synchronize: All new and changed files from source are copied over to destination. Deleted files on source get deleted on destination, but any changes on destination do not get changed on source.
Mirror: All new, changed and deleted files on either source or destination are changed on the other drive.
I almost always use Synchronize. So when I load new image files on source, synchronize copies them to destination. If I import them in darktable and make some edits, then run Synchronize again, only the xmp files are copied since those are the only files new or changed. If I edit in Capture One, only the catalogue and some other text files are copied when I Synchronize.
I do use one external portable drive and while on the road will edit photos - for this I use Mirror since I want the destination location (the portable drive) to copy changes over to the source drive (my hard drive).
Many people use full Backup mode…they like the fact that nothing gets deleted. But for me I’m ruthless on culling photos - if I delete, they are gone. Restoring from a full backup would be a nightmare if you don’t use starring or colour codes properly to easily see what you’ve deleted in the past.
I’m on Windows, and as someone else mentioned, I also used FreeFileSync at one point, but now only use SyncBackFree. Let’s me set up multiple different “profiles” that I choose to run when I want and depending on the situation.