Why does the water in the toilet move around so much on stormy days?

Any siphon in Your house will be affected in this way. The inside surface (that you can see) is affected by the air pressure in Your house, the other level is connected to the sewer piping system, wich gets his pressure from a vent of the sewer system.

Both pressures can be affected in many ways by the stormy wind, depending where some passages (like windows not really tight) for the air are located , eg at the windward or leeward side of Your house. The vent of the sewer system can develop pressure variations depending on wind speed by some water-aspirator-like action.

So the siphon in Your toilet works as a pressure differential indicator.

Sewage drains in houses should always have a "vent stack" - piping which extends up through the roof. The vent portion of the sewage plumbing has two purposes: to allow gases in the system to escape, and to provide a "suction break" so that traps (U-bends) don't have their contents sucked out when the pipe fills with flowing water (the water in a trap is there to keep sewage gases from entering your house).

So, there is a pipe sticking up through your roof which connects to the smelly side of your drain plumbing, and it connects (eventually) to your toilet. While there is no water flowing down the drain, the only water in all that piping is just enough to fill the trap.

When wind blows across that vent pipe on your roof, it lowers the pressure inside the pipe (Bernoulli principle). The difference in pressure that creates relative to the air pressure inside your house causes some of the water to be drawn toward the "smelly" side of the trap and you see the level drop. When the wind abates, the pressures equalize and so do the water levels in the trap. Stormy weather produces gusts, so there will be rapidly fluctuating winds across the vent pipe, so rapidly changing pressures inside it, so the water level in the trap dances up and down.