I think part of the name confusion stems from the fact that today's "mirrorless" cameras are a fairly recent introduction in digital photography. Where we once were able to distinguish point-and-shoot cameras from DSLR cameras pretty easily, the trend in the industry has been to blur those line more and more.
So-called "bridge" cameras might be seen as the first salvo here, offering improved optics and features (compared to traditional P&S cameras) without making the jump to multiple lenses and (in most cases) large sensors.
The term "mirrorless" is most frequently applied today to Mirrorless Interchangeable Lens Camera (MILC) systems or Electronic Viewfinder Interchangeable Lens (EVIL) systems. In these cases, the two distinguishing characteristics are the lack of a moving DSLR-style mirror and the ability to change lenses. Sony's translucent mirror technology might be considered a fringe member of this class, though I believe its feature set and target audience, plus the fact that it does, in fact, have a mirror, probably place it closer to traditional DSLRs.
A final consideration for these cameras is the sensor. Although mechanically, these cameras resemble P&S cameras, many of them feature sensors more closely resembling (or identical to) crop-sensor DSLRs, vs. the typically smaller sensors found in P&S cameras. Again, you can't draw perfectly clear lines here, since there are some P&S cameras with larger sensors (Canon's G-series comes to mind). In some ways, in fact, these could be seen as predecessors or inspiration for some of today's mirrorless offerings.