MacOS – Rsync with Linux server: special character problem

I have a problem with using rsync for backuping files on my remote linux machine to my Mac. The first time all is well, but apparently there is a problem with special characters between the mac and linux machine, since every time I run the rsync operation after that, the files with special characters are first deleted and then re-synced. It seems as if there is a problem with different character sets.

The preferred solution seems to be to use the --iconv option:

You can use rsync's –iconv option to convert between UTF-8 NFC & NFD,
at least if you're on a Mac. There is a special utf-8-mac character
set that stands for UTF-8 NFD. So to copy files from your Mac to your
NAS, you'd need to run something like:

rsync -a --iconv=utf-8-mac,utf-8 localdir/ mynas:remotedir/

This will convert all the local filenames from UTF-8 NFD to UTF-8 NFC on the
remote server. The files' contents won't be affected.

That sounds all and well, but my mac does not seem to recognise that option, since I get:

>     MyMachine:~ Macuser$ /usr/bin/rsync -av --delete --iconv=utf-8,utf-8-mac user@linuxmachine:/home/linuxuser/Test/ /Users/Macuser/Test/
>     rsync: --iconv=utf-8,utf-8-mac: unknown option
>     rsync error: syntax or usage error (code 1) at /SourceCache/rsync/rsync-42/rsync/main.c(1333) [client=2.6.9]

Any suggestions on what to do?

EDIT: the rsync version on my mac is 2.6.9.

Macuser$ rsync --version
rsync  version 2.6.9  protocol version 29

If that version does not support the --iconv option, how do I update in a safe manner?

EDIT: I finally figured out that apart from updating rsync, I also had to switch the order in which I specified the character sets, so I added an answer below.