These two terms get thrown around a lot, by me most of all, while what they really mean is very much open to interpretation. I do not have the power to unilaterally define them. I can, however, at least explain what I mean when I use them.

The term crossover scope arguably came about due to the advent of western hunting. Out west, it is not uncommon to take a shot on game from a considerable distance often in pretty gnarly conditions. That requires something akin to a precision scope if you want to do it well. Modern precision oriented riflescopes are frequently large and heavy. While it is possible to drag one uphill with you, it is not something I recommend. Plenty of people do it and I applaud their mental fortitude. I've done it and I don't like it. It is hard enough to drag my fat ass uphill. I am clearly not the only one, since quite a few years ago Premier Reticles came out with their Light Hunter scope. That was a version of their 3-15x50 design, except with simpler and lighter turrets all stuffed into a 30mm tube. Optically, it was the same as the larger tubed scope, but it weighed a lot less. It gave up some adjustment range and turret features, but that's largely it. When Premier went under Armament ended up with their assets. The scopes were re-engineered and improved. Then re-designed again and merged with a different ongoing development project that resulted in Tangent Theta scopes. TT315M and TT315H are the spiritual successors to the Premier Light Hunter: 3-15x50 scopes with 30mm tubes, limited adjustment range, simpler turrets and spectacular optical performance. They weigh around 27 ounces.
In parallel, Bushnell made their 3-12x44 and 4.5-18x44 LRHT scopes for a while (eventually they Bushnell-ed them), Leupold developed the 3.6-18x44 Mark 5HD and, more recently, Vortex introduced a 4.5-22x50 Razor HD-LHT. The latter, along with the Tangent are probably the purest expressions of the crossover concept we have today.
What's a crossover scope, then? To me, it is a FFP scope with max magnification of no less than 15x, illuminated mrad (or MOA for the masochists among us) reticle, elevation turret equipped with a zero stop and locking or covered windage turret. The objective lens diameter should be between 40 and 50mm. The scope weight should be no more than 30 ounces. Many of these criteria are somewhat arbitrary, but I have to draw the lines somewhere. On the low end I am looking for magnification of not more than 5x for an occasional offhand or badly supported shot. It is important that the reticle works adequately well across the entire magnification range, which is one of the reasons I require illumination. That is also why, to me, tree or grid type reticles that illuminate the whole thing are a problem. I want a small center portion of the reticle illuminated: a dot or a crosshair. You can get clever with it and illuminate a few more hold points, but not the whole thing. That has a tendency to make the scope useless in low light.

MPVO stands for Medium Power Variable Optic and the concept originated when people started thinking "what if I wanted to make an LPVO like scope, but did not need 1x on the low end?" 1x definitely complicates everything and one thing we can do by getting rid of it is bump up the objective diameter a little and optimize optics better for mid-range magnification. Original MPVO concept was sort of like a "tactical tweener". The purest expression of that original idea is Leupold Mark 5 2-10x30, Vortex PST Gen2 2-10x32 also sorta barks up the same tree, but it really did not set the world on fire. The Mark 5HD is an interesting design that I plan to test in the not too distant future. Given the absolute reluctance of most manufacturers to make high quality FFP riflescopes with objective lens diameters between 30 and 40mm, the definition of an MPVO got extended to include larger designs with 40-ish millimeter objectives. For these, the definition is a little different in that one of the binding criteria is low magnification, with higher magnification being a lot more flexible. To me, one of the absolutely critical requirements of an MPVO is reasonable compatibility with commercially available thermal clip-ons. That means the low end FOV had to be at least 7.5 degrees or thereabouts. That translates to around 40ft @ 100 yards. Wider is even better (closer to 50), but 40 is reasonable. For most riflescopes, that's between 2.5x and 3x. For some short-ish eyerelief prismatics, you can get close to that FOV on 4x, but they are not variable optics.
The rest of the specs are simple: FFP reticle with some ability to hold over, locking or covered elevation turret preferably with zero stop (but reticle only distance and wind holds are acceptable, so no zerostop is OK), covered or locking windage turret, objective lens diameter between 30 and 44mm, weight of less than 30ounces. Lighter is naturally better.
There is some obvious overlap between MPVO and crossover optics, but I prioritize different features with these two scope types.
Here are the two configurations I would consider ideal that, hopefully, illustrate my point.

Ideal crossover:
30mm tube, illuminated tree reticle (Vortex XLR2 or similar), 3.4-17x50, side focus, 20 ounce weight, 3 - 3.5" eye relief, single turn 12mrad turret with 0.1mrad clicks, no low power tunneling, covered low profile windage turret, exposed low profile zerostop elevation turret. Essentially, I am looking for a fusion of Razor HD-LHT and USO FDN-17x.

Ideal MPVO:
30mm tube, illuminated abbreviated tree reticle (S&B's MDR-T6 or similar), 2.4-12x36, side focus, 20 ounce weight, 2.5 - 3" eyerelief, single or double turn turret with 0.1mrad clicks and at least 10mrad of available adjustment, locking zerostop elevation turret, covered low profile windage turret. Essentially, I am looking for a fusion of the Mark5 with Blackhound Emerge or something in between. Leupold turrets are near perfect for this application.

Neither configuration currently exists. Both are eminently makeable and whichever OEM currently makes the Razor HD-LHT 4.5-22x50 with sub 22 ounce weight, can pull off these exact configurations if properly motivated.

Unfortunately, there is no one out there except for me who thinks these will sell. The wait continues....