In that case your monitor, keyboard and mouse are operating systems.

You're confusing OS with window manager or command prompt.

The filesystem is the OS part of DOS.

AFAIK DOS doesn't have memory management.

I believe DOS has a memory allocation function related to its real mode segmented memory plan but is completely incapable of utilising a memory management unit - even just the segmented mmu on the 286.

List of DOS services.

That's why I said AFAIK

What are KDE, Gnome, *box on linux? Part of the OS or just addons?

Whenever you integrate them to the system they become part of the OS.

And according to the link i've posted above. Any command prompt or shell
or whatever you want to call your 'low level interface'
(I don't know, but i think the ability to type in text through a keyboard
and directly seeing characters appear on some sort of display is pretty
high level...)
is also considered part of the OS.

[edit]
Or to express it another way:
kernel and shell(shell used as a synonyme for user interface)
might be two seperate parts, but only a combination of any two compatible
kernel and shell would make me look at it as an OS.
Even on machines that are not personal computers (like a washing machine)
there needs to be some buttons and lamps to make the 'shell' so
that together with the embedded kernel the washing machine has its OS.

The DOS and unix command prompt/shell, and the windows window manager can be seen as part of the OS. But you were saying that's all an OS is. Anyway, when I'm talking about OS in this thread I'm refering only to the kernel.

Common to who? Whole unix/linux world sees linux kernel as the OS, all the other things are just optional addons

to run old programs that no one updates anymore

The only reason why DOS or DOS-like systems are hanging on is because they are very simplistic in comparison to other operating systems. Some are also royalty free. They are normally used for low level hardware interaction (flashing bios, running diagnostics tools), and very little else. With more complex operating systems, things can get in the way and cause problems when all you want to do is something very basic. Under DOS, all this overhead is removed and you can be certain that the end user meets the software requirements.

DOS is considered "dead" in my books, because it is not something that you would expect to find running on an end user's system. It is something that you need the end user to invoke by means of a start-up disk. Having said that, everyday tools are not developed enough for DOS in order to consider it "alive"
Word processors, calculators, media applications, data managers, telnet clients, or other utilities that many people use on a daily basis either
1) Cannot be found.
2) Have not been updated in years.
3) Have a severe lack of functionality in comparison to software for other OS's

You guys are getting too picky about the definition
There are a hundred ways to say what an OS is. Here's my one-liner.

An OS is a software based system designed to provide a method of using the functionality of the hardware it is installed on.