This is a late answer; a recent question was marked as a duplicate of this.
The phenomenon discussed in the question goes under the general concept of the dynamics of sloshing liquid. Sloshing liquids can overturn tank trucks, derail railroad tanker cars, capsize ships at sea, crash aircraft, and cause spacecraft to lose controllability. This makes this a very important concept for economic and safety reasons and hence is the subject of many journal articles and entire technical books.
The dynamics of slosh are nonlinear, rather complex (particularly so if the sloshing is extreme and creates bubbles), and are highly dependent on container geometry. As a general rule, a container that is nearly full or nearly empty of fluid doesn't slosh much, and sloshing is at its worst when the container is close to half full.
Excitations from vehicle suspension, from vehicle acceleration and braking, from ocean waves, and from the control systems of aircraft and spacecraft can turn low amplitude sloshing into high amplitude sloshing, something that is best avoided.
But I'd like to see a mathematical explanation of this phenomenon.
You are inadvertently asking me to write a lot. Go to scholar.google.com and books.google.com and search for "slosh dynamics" and you'll see how much has been written on understanding and mitigating slosh. I'll instead provide an overview.
Most slosh models are a bit ad hoc. A simple approach is to model the fluid as being partitioned into a fixed part (one that moves with the container) and a sloshing part, with the sloshing part modeled as a spring/mass/damper system or as a damped pendulum system. The slosh wave slams into the container wall, and this has to be modeled as well. This works well for low amplitude slosh, not so well for high amplitude slosh. These low amplitude slosh models yield a natural slosh frequency. These models predate modern computing.
More recently, slosh has been studied using computational fluid dynamics, and even more recently, with smoothed-particle hydrodynamics originally developed by astrophysicists to model galaxy formation, star formation, supernovae, etc. The same techniques work quite nicely to model more mundane fluids such as sloshing in a container.
In the cases of Bottle A and Bottle C, they are full and empty. So the amount of water in them (or not) can be considered to be a complete system along with the bottle, since there is no possible way in which the fluid in the full bottle would reduce its volume or overall distribution, certain properties like the system's center of mass,center of gravity and its overall mass distribution stays constant.
But considering bottle B, when you try to flip it, the centrifugal force created by the force you apply causes the quantity of water in the bottle to shift, which causes a shift in its center of gravity, and a change in its mass distribution, which can cause an overall change in the bottles rotation.( The moment of inertia of the body changes. The moment of inertia is a property which is analogous to mass, which is used to study rotation. A body with more moment of inertia would rotate less freely and vice versa). So it indeed is because of the quantity of the water in the bottles.
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This is why you don't want to ship oil across the ocean in a half-full oil tanker. If you do, you had better equip that tanker with some very good anti-slosh mechanisms. The same goes for trucks, trains, and spacecraft carrying fluid. One quarter to three quarters full is when fluid sloshing (linear fluid dynamics), slamming (non-linear fluid dynamics), and whipping (highly non-linear fluid dynamics) are at their worst. Above three quarters full, most of the fluid cannot participate in the sloshing, slamming, or whipping. Below a quarter full, most of the fluid is free to participate in the sloshing, slamming, or whipping, but there's not much fluid in the container.
In the case of the bottle full of water, there's no room for the fluid to slosh or splash. The angular momentum you impart to the bottle is quickly transferred to the fluid. The bottle and fluid rotate as one.
In the case of the half-full bottle of water, you aren't immediately transferring angular momentum to the water. Instead, your initial flip initiates a slosh wave. That slosh wave is large in amplitude and doesn't have to travel far before it hits the other side of the bottle. This is non-linear dynamics. That wave smashing into the other side of the bottle marks when a good deal of angular momentum is transferred to the water. The bottle's rotation rate slows down markedly at this point (but it does not come to a stop).
In the case of the nearly empty bottle of water, the transfer of angular momentum to the water once again isn't immediate. Once the transfer has been complete, the bottle rotates about a point well below the center of the bottle. Given the reduced mass of the water and the lowered center of rotation, a good share of the angular momentum remains with the plastic bottle rather than being transferred to the water. The bottle rotates faster than is the case with the half empty bottle.
I can't fully replicate your results. I can give a full bottle of water a very hefty rotation rate by imparting some backspin while I toss the bottle. I couldn't make a partially filled bottle rotate anywhere near that fast. The half-full bottle wants to come to a near stop mid-flight unless I crank my arm around a few times before letting go. (This lets the slosh wave hit the other side prior to release.) The near empty bottle does rotate faster than the half-full bottle, but not as fast as the plumb full bottle.