While others have given you examples of how to get this done in ID…
In JOSM, this really is just a simple Alt+X
after you install the helpful “UtilsPlugin2” JOSM plugin!
See an explanation+GIF of it in action in:
How to Split a Building in JOSM
0. Press A
to get into “Draw” mode.
1. Left-Click on one edge/node of the building.
2. Move the mouse and Left-Click on other edge/node of the building.
3. Press Alt+X
.
- Or More Tools > Split Object.
Done. 
And repeat 1–3 as many times as needed.
(To get this set up in JOSM, see the 2 tutorials I posted below.)
Side Note: I first began using it after manually splitting/“redrawing” many long rectangles of townhouses (or duplexes) into their individual houses.
(And this method really is so much better/faster than that current slow/clunky ID method.)
Now, I even have my extra mouse button set to Alt+X
, so all I have to do is:
- Draw line.
- Click side-mouse button.
So I can split apart buildings as fast as I can do 3 clicks! 
How to install UtilsPlugin2
in JOSM
1. Edit > Preferences
2. Go to the “Plugins” tab.
3. Type utils
into the search box.
4. Check the box ON next to utilsplugin2
.
- (As of today, it’s version 36362.)
5. Press OK button.
Now, close out of JOSM and reopen it from scratch.
Note: In order to more easily edit/open the location in JOSM, I also set up “Remote Control”.
This lets you easily just:
- Visit the location on OSM.org.
- For example, clicking on your forum link in Post #1.
- You see that Green “Edit” button up top?
- Click on the little dropdown next to it.
- Press “Edit with Remote Control (JOSM, Potlatch, Merkaartor)”.
Now it’ll pop open that location in JOSM. You can then:
- Do your edits in JOSM
- Push the button to upload
and it’ll send all your changes back to OSM. 
How to set up “Remote Control” in JOSM
1. Edit > Preferences
2. Go to “Remote Control” tab.
3. Check the box ON for “Enable remote control”.
Then restart JOSM.
Now, all you have to do is:
- Have JOSM open when you push that “Edit with Remote Control” green-button-dropdown
and it’ll automatically transfer the info to JOSM and load up that spot of the map.
Happy mapping! 