Running games in windowed mode with black borders
Solution 1:
To answer your question: no. Not a "good" way. Borderless mode, or Fullscreen Windowed mode as you call it, will fun the game as "fullscreen", though as a borderless window application. said window is not re-sizable. An adequate solution would be to open a completely black picture to have underneath a regularly windowed game, and set the resolution on said game lower. You will have the border from the game, and you'll have to open a picture every time you wanna game though (plus, it's another thing to have to keep in front of your browser and such after browsing something). Personally, I have yet to play a game that allows resizing of the rendered area in the game window at all, much less re-sizable borderless windows.
Solution 2:
Black borders will not be applied automatically (excepting perhaps some weird, game-specific support, but don't count on that). The closest thing if your graphics card or monitor support it is to run the game monitor at 1920x1080 and tell it to use black bars instead of stretching to fit the video feed to the display size. You'll just have to change the size back and forth each time.
You can use an auto-hotkey script to size and position game windows running in windowed borderless mode, and then try to find another program which will function as basically a giant black box that covers the desktop. This should also get you pretty close to what you want.