Nowadays, there are true "point and shoot" cameras which operate almost exclusively in Auto type modes with virtually no user control, middle models which have perfectly capable auto modes but also allow for more user control, more optical reach options, etc., and advanced P&S models which can have very good manual control-ability, excellent lenses, and large sensors. Categorically, I still call all of these 'P&S'.
Just as 'mirrorless' has pretty well come to be the catch-all phrase meant for interchangeable lens cameras with electronic finders or LCDs and larger sensors. Taken literally, any camera that is not a DSLR is technically 'mirrorless', which would include all fixed lens and compact small sensor 'P&S' models...but the term 'mirrorless' or MILC has become accepted terminology for the interchangeable lens systems.
And even in the DSLR world - I personally always stuck Sony's 'SLT' models in the DSLR category, even though technically they weren't. Again, the classification wasn't meant literally, but just as a familiar category for large sensor, interchangeable lens camera systems which were generally used for traditional lens mounts with larger registration distances and some kind of mirror and dedicated focus sensors. Sony was still using A-mount, which started as an SLR mount, and the translucent mirror didn't move but was still a mirror, serving a dedicated PDAF sensor.
Anyway, that's how I'd break it down. RX, HX, TX, WX, and so on - all P&S. NEX, A7xx, A9xx, A6xxx, etc all mirrorless, and all Nikon, Sony, Pentax, & Sony traditional SLR mounts in the DSLR category.