I honestly think it was always meant to be a bit ambiguous and really just a future sci-fi play on "the demilitarized zone."
I suppose it's just supposed to be an area of space between two territories that neither faction can enter and act as a buffer between them. So they don't take pot shots over the fence, if you will.
In 1922, Saudi Arabia and Iraq couldn't make up their minds where the border between their countries should be. So a small area ("the border is somewhere in there" area) was created. (A smaller one was also created at the Saudi Arabia and Kuwait "border.") As spelled out in the Protocol of Uqair, no government could build fortifications or station troops in or near the zones, or near the border generally--but nomadic tribes (who pretty much decided that sand is sand) could come and go as they please.
The actual "neutral zones" were dismantled in 1970. (Everyone pretty much agreed on the borders by then.)
My hunch is that the Romulan Neutral Zone and/or the Klingon Neutral Zone derive conceptually from those actual neutral zones.