Thanks for reading.
The invariant interval between two events is defined as...
...where $t$ is the time between the events and $x$ is the distance between the events.
When its positive, that means that we can find an observer that's at the location of the first event and at the location of the second event, just as the events happen. In that case, $S$ would be $c \tau$, the proper time that passes for that observer between the two events multiplied by the velocity of light, aka the distance that a light-beam shot by him just as the first event happens would travel.
In case the above gives trouble, the following video explains it pretty well:
However, some books define the invariant interval like so:
This invariant interval is positive when the other is negative...but, what does it represent? It can no longer be interpreted as proper time...so, what's its interpretation?
I can see that it gives equations of horizontal hyperbolas in a Minkowski Diagram...but, I'm not sure how to interpret those (as for the vertical hyperbolas, I know to interpret them as each joining all events of constant proper time for an observer moving relative to the stationary frame in the diagram).
Is it...proper length? Length between events for the "stationary" observer? Hmm...can we make a similar light-experiment that shows what it is, the same way the one in the linked video shows how the invariant interval can be the proper time (multiplied by the speed of light)?