After a long time, I like to bring that thread to an end.
Since I also had some other trouble when trying to copy a larger amount of data, I recognized by chance (and with some help of a âLinux-expieriencedâ friend of mine) that the copy procedure ran into some memory violation, which led me to the suspect of system memory failure.
Due to this, I started some extended memory checks with memtest86+; and indeed, the system ran into trouble again and again after the second, third, fourth (âŚ) pass of the complete check, every time when accessing the higher memory addresses (b.t.w., most times, it managed the first run without any hazzle).
So I tried a lot of things, cleaned the contacts, swapped the memory modules between the slots, and tried some other RAM modules and combinations of different modules in both slots; in the end, I found a combination that seemed to work stable even after fife or six complete runs in memtest86+ (at the expense of capacity + 10MB instead of 16MB and dual-channel operation).
After this measure, I re-installed the (original) printer driver, and since then, the printing seems to work stable and reliable.
So I think that the problem was indeed caused by the memory failures: I guess that when being logged in as the only user in parallel, the lower memory was sufficient, but when switching to another user in parallel (without being logged out before), the processor also started to access some memory at higher addresses, which caused memory failures and crashed the printing server somehow.
In between, I bought a new laptop (Tuxedo InfinityBook Gen9, AMD flavor) and also installed Manjaro on it (Iâm currently using it to write this answer
), and yesterday I added the Brother Printer to it, which was immediately recognized correctly, and the correct printer driver was automatically used. 
Although the Manjaro system thatâs running on the old Dell (which now is mainly used by my mother in law) seems to work well these days, Iâm thinking about to re-install a fresh Manjaro and checking if it now works as flawless as it does on my new Tuxedo.
After this, I will write the outcome here to complete the information and close this thread.
Anyway, a big thanks to everyone supporting me here, especially @linux-aarhus, @jrichard326 and @Deano.