RESOLVED!!! Yay!
Here's what I did, perhaps someone else will find it useful.
1. In Terminal, type:
ioreg -lw0 | grep IODisplayPrefsKey
2. You will get a string ending with "... AppleDisplay-xxxx-yyyy" (numbers at the end will vary).
3. In Finder, go to
/System/Library/Displays/Overrides/
Find DisplayVendorIDxxxx folder
See if this folder has a DisplayProductIDyyyy file
If the file is there, you might want to copy it to another location for future use.
If there is no file with that name (like it was in my case):
4. Copy another file (I just used the largest file from DisplayVendorID610 (Apple), namely DisplayProductID-9cd7, you may want to try something else), paste it into DisplayVendorIDxxx folder and rename to DisplayProductIDyyyy
UPDATE: Another (and likely more proper) solution would be to use the display profile from the "terminal" Mac (in my case, iMac 2008) by running ioreg -lw0 | grep IODisplayPrefsKey command there.
5. Reboot.
After reboot, I have an iMac display in Display properties, and access to all resolutions up to 1920x1200. I suppose you can dig deeper if you like and find something larger too, but this works good enough for me.
Disclaimer: The display I use as a dummy is an old, broken, 15-inch ViewSonic with no AC adapter (yes, not plugged in, attached to Mac Mini via DVI cable). I have no clue what this solution does to a live monitor, so use at your own risk.