Space-like separated events are events that, in a well-chosen reference frame, can take place at the same time but never happen at the same location.
On the other hand for time-like events, one can chose a reference frame such that they happen at the same place but never simultaneously.
I can't help thinking that the labels are therefore very ill-chosen... Is there another motivation for these names?
Suppose you draw a spacetime diagram and measure the angle of your trajectory to the spatial and time axes. You'll find time like trajectories are more nearly parallel to the time axis - i.e. the angle the trajectory makes to the time axis is smaller than the angle it makes to the spatial axis. Conversely space like trajectories are more nearly parallel to the spatial axis. This seems to me a plausible origin for the terms.
I admit that the terminology is not as self-explanatory as it should, however a source of confusion is the fact that you are actually looking at consequences of the definitions, instead of at the original definitions that hold in every connected time oriented spacetime. The terminology turns out to be more clear if you use the original definitions which are referring to the nature of the curves connecting the events.
For a pair of events in a generic spacetime, time-like related means, by definition, that there is a future directed time-like curve joining the points. In Minkowski spacetime, it is equivalent to say that there is a time-like geodesic joining the events and it implies (it is equivalent in that spacetime) that there is a Minkowski reference frame where the events have the same location at different times.
For a pair of events in a generic spacetime, causally related means, by definition, that there is a future directed causal curve joining the events. Causal curve means that its tangent vector is not space-like. Causal curves are those curves describing the stories of physical points transmitting interactions.
Finally, for a pair of events in a generic spacetime, space-like separated (or also, equivalently, causally separated) means, by definition that there is no future directed causal curve joining the events. In Minkowski spacetime, it is equivalent to say (and it justifies the name) that there is a space-like geodesic joining the events and, in turns, it implies (it is equivalent in Minkowski spacetime) that there is a Minkowski reference frame where both events occur at the same time in the rest frame of the reference frame.
Think about it this way: Any event outside of another event's light cones is called "space-like separated" from that event because spatial separation dominates the difference.
For example, imagine two events that lie on what the wiki-link picture calls the "hypersurface of the present": these two events are causally disconnected because they are too far apart for light to reach one from the other (despite appearing to occur at the same time).
On the other hand, events which lie inside each other's light cones are "time-like separated" because temporal separation dominates the difference (i.e. one will always be in the other's past regardless of reference frame). Moreover, if two events are time-like separated they are well-ordered (one always comes before the other), but if they are space-like separated, there exist reference frames in which either one you choose happens first.
The terminology makes perfect sense if you include the word interval after space-like and time-like. If two events can never occur in the same place, regardless of which reference frame you choose to view them in, it seems reasonable to assert that they are separated by a space-like interval. Similarly, if one event must always occur after the other, it makes sense to say that they are separated by a time-like interval.
The important point is that the space-time interval is invariant under transformations between reference frames (Lorentz transformations). Therefore, we can meaningfully talk about different classes of intervals (space-like, time-like, and light-like). Humans are not great at visualizing distances in four-dimensional space-time, so these names are valuable in developing intuition.
