Consider two infinitesimally close, timelike separated but otherwise arbitrary events $P$ and $Q$ with coordinates $(t,\vec{x})$ and $(t+dt,\vec{x}+d\vec{x})$. For example, imagine event $P$ is "a bulb turns on in a room" and the later event $Q$ is "someone sneezes in the room". The invariant spacetime interval between them is $$(ds)^2=-c^2(dt)^2+|\vec{dx}|^2.$$
Is it possible to write the timelike interval between any two such random events as $$(ds)^2=- c^2(d\tau)^2?$$ If so, can we interpret this $d\tau$ as some proper time interval? But proper time interval of what? Proper time is usually linked with a moving object/observer; it is the time recorded by a moving object/observer in its rest frame.
In this scenario, however, we do not have a moving object to start with. But we can imagine a moving observer who has the correct velocity so that at time $\tau$ in her clock, she is at the spatial location of $P$ (location of the bulb) and at the time $\tau+d\tau$, at the spatial location of $Q$ (location of the sneeze). Thus, the $d\tau$ of this case, can be thought of as the proper time interval of a special observer who has an appropriate velocity to be at the location of the events when they occur. Am I correct?