O.K., I did now what you suggested. Thanks so much @Akito 
● fwupd-refresh.service - Refresh fwupd metadata and update motd
Loaded: loaded (/lib/systemd/system/fwupd-refresh.service; static; vendor preset: disable>
Drop-In: /etc/systemd/system/fwupd-refresh.service.d
└─override.conf
Active: inactive (dead) since Fri 2022-09-23 15:42:27 CEST; 38ms ago
TriggeredBy: ● fwupd-refresh.timer
Docs: man:fwupdmgr(1)
man:fwupdmgr(1)
Process: 18679 ExecStart=/usr/bin/fwupdmgr refresh (code=exited, status=2)
Process: 18695 ExecStart=/usr/bin/fwupdmgr refresh (code=exited, status=2)
Main PID: 18695 (code=exited, status=2)
Sep 23 15:42:27 rosika-10159 systemd[1]: Starting Refresh fwupd metadata and update motd...
Sep 23 15:42:27 rosika-10159 systemd[18679]: Found pre-existing private CacheDirectory= direct>
Sep 23 15:42:27 rosika-10159 systemd[18679]: Apparently, service previously had DynamicUser= t>
Sep 23 15:42:27 rosika-10159 systemd[1]: fwupd-refresh.service: Succeeded.
Sep 23 15:42:27 rosika-10159 systemd[1]: Finished Refresh fwupd metadata and update motd.
Well, it looks a bit better now, I think.
Active: inactive (dead) since Fri 2022-09-23 15:42:27 CEST; 38ms ago
… this line doesn´t say “failed” anymore, but rather “inactive (dead)”.
Then:
instead of
Sep 23 14:19:52 rosika-10159 systemd[1]: fwupd-refresh.service: Failed with result 'exit-code'.
I get
Sep 23 15:42:27 rosika-10159 systemd[1]: fwupd-refresh.service: Succeeded. 
But here :
Process: 18679 ExecStart=/usr/bin/fwupdmgr refresh (code=exited, status=2)
Process: 18695 ExecStart=/usr/bin/fwupdmgr refresh (code=exited, status=2)
the contents within brackets are printed red. Don´t know what to make of that.
And: the extit status of the last command is:
echo $status
3
Hmm. is everything now as it should be 
I really don´t know…
But thank you so much for your help @Akito
Many greetings from Rosika 