I don't see any reason to distinguish between “monitor” and “screen” the way you describe. In fact, I wonder where you got this distinction; I've always seen the terms used interchangeably, except inasmuch “screen” is sometimes the part of the monitor that displays the picture (inside the frame). If you want a neutral term, you could use “display” — but for a tag name, you really want to capture common terms, even if they aren't “proper”.
On the Unix site, we use
screen for the application and
monitor for the hardware device. On Super User,
gnu-screen is for the application and
monitor seem to be used interchangeably (excepts that
screen inevitably attracts questions about the application).
For what it's worth, outside Stack Exchange, I'm not used to seeing Screen referred to as GNU Screen, so until I saw it on Super User it wouldn't have crossed my mind that a question about Screen should be tagged
gnu-screen and not
screen. Of course, I do realize that many people wouldn't know about Screen and would naturally post monitor questions with the
screen tag. (But they ask monitoring questions with