and the tutorial I've seen told me to use sys_sync syscall
Well that sounds silly, unless you're doing filesystem write benchmarking or something.
If you have one really critical file that you want to make sure is "durable" wrt. power outages before you do something else (like sent a network packet to acknowledge a complete transfer), use fsync(fd) to sync just that one file's data and metadata.
(In asm, call number SYS_fsync from sys/syscall.h, with the file descriptor as the first register arg.)
But my problem is why and when should we use this?
Generally never use the sync system call in programs you're writing.
There are interactive use-cases where you'd normally use the wrapper command of the same name, sync(1). e.g. with removable media, to get the kernel started doing write-back now, so unmount will take less time once you finish typing it. Or for some benchmarking use-cases.
The system shutdown scripts may run sync after unmounting filesystems (and remounting / read-only), before making a reboot(2) system call.
Re: why sync(2) exists
No, your data isn't already on disk right after echo foo > bar.txt.
Most OSes, including Linux, do write-back caching, not write-through, for file writes.
You don't want write() system calls to wait for an actual magnetic disk when there's free RAM, because the traditional way to do I/O is synchronous so simple single-threaded programs wouldn't be able to do anything else (like reading more data or computing anything) while waiting for write() to return. Blocking for ~10 ms on every write system call would be disastrous; that's as long as a whole scheduler timeslice. (It would still be bad even with SSDs, but of course OSes were designed before SSDs were a thing.) Even just queueing up the DMA would be slow, especially for small file writes that aren't a whole number of aligned sectors, so even letting the disk's own write-back write caching work wouldn't be good enough.
Therefore, file writes do create "dirty" pages of kernel buffers that haven't yet been sent to the disk. Sometimes we can even avoid the IO entirely, e.g. for tmp files that get deleted before anything triggers write-back. On Linux, dirty_writeback_centisecs defaults to 1500 (15 seconds) before the kernel starts write-back, unless it's running low on free pages. (Heuristics for what "low" means use other tunable values).
If you really want writes to flush to disk immediately and wait for data to be on disk, mount with -o sync. Or for one program, have it use open(O_SYNC) or O_DSYNC (for just the data, not metadata like timestamps).
See Are file reads served from dirtied pages in the page cache?
There are other advantages to write-back, including delayed allocation even at the filesystem level. The FS can wait until it knows how big the file will be before even deciding where to put it, allowing better decisions that reduce fragmentation. e.g. a small file can go into a gap that would have been a bad place to start a potentially-large file. (It just have to reserve space to make sure it can put it somewhere.) XFS was one of the first filesystems to do "lazy" delayed allocation, and ext4 has also had the feature for a while.