Water siphons have been demonstrated to 24 meters. Water can resist -280 atmospheres pressure without vaporizing. Corresponding to possible siphon heights of more than 2800 meters. Siphons can operate in a vacuum. Siphoning of mercury has been demonstrated to more than 30cm above the barometric height, even in glass which mercury adheres poorly to.
24 meter siphon:
Siphon of ionic liquid in vacuum:
Siphon of mercury to 30cm above barometric height:
Negative pressures of -280 atmospheres in water have been demonstrated in the ingenious Z-tube:
The Z-tube is a z-shaped tube nearly filled with liquid and set on a spinning table. If the liquid starts to shift away from the center, the "height" of the liquid in the bent inward ends "rises" toward the center, increasing pressure in that end and returning the liquid to the center. By measuring the spinning speed and the distance from the center to the liquid level in the ends, the pressure can be calculated. It helps if the tube is of a material the liquid will adhere well to. And the tube must be very clean and the liquid degassed to prevent cavitation.
Another example of negative pressures in water are in the xylem of very tall trees. The water does not rise by capilary action very far. The water is pulled up by action in the leaves at top. Negative pressures of several atmospheres are achieved in tall trees.
So, many people are correct that liquid cohesion DOES pull the liquid over the top of a siphon in SOME siphons. And everyone agrees that all siphons rely on gravity (or similar acceleration) for their effect. But most practical siphons don't rely on liquid cohesion. And some siphons CAN'T use liquid cohesion to pull the liquid over. It is not the case that only one of the theories: atmospheric pressure, gravity, or liquid cohesion, is the answer to how a siphon works. All three of those explanations are involved. We don't have to choose just one.
One example is the siphoning of CO2 gas, which has been demonstrated. And a demonstration you can easily do with a garden hose is like figure 4 of the Wikipedia siphon article, fill the tall down side of a siphon with water, but leave the top and short up side with only air. When the water in the tall down leg is released, gravity will reduce the pressure at the top of the siphon and atmospheric pressure will push the water from the upper reservoir up and over the siphon. Since the water on each side of the siphon is not touching at the start of this experiment, liquid cohesion cannot explain what force raises the water. The air at the top of the siphon, though reduced in pressure, is still at positive pressure relative to complete vacuum, and therefore it is trying to expand, and pushing DOWN on BOTH sides of the siphon. Since gravity is also pulling down, only atmospheric pressure can supply the force to push the liquid up into the low pressure zone created at the top of the siphon by gravity pulling down the liquid in the taller down tube.
Another observation of the difference between vacuum siphons and practical siphons is that in practical siphons, small and even fairly large air bubbles can flow over the siphon without much change in its working. Whereas in a vacuum siphon, a bubble or void will immediately expand to break the siphon.
In practical siphons near sea level, liquid cohesion is not only unnecessary, it cant even contribute, because all the fluids in the siphon are at positive pressure relative to complete vacuum and therefore all the molecules are being squished together and are repelling each other. There can be no pulling in siphons near sea level pressure. Atmospheric pressure pushes the liquid up despite that the pressurized fluid at top is trying to push the fluid DOWN on both sides.
Atmospheric pressure pushes the liquid up in a typicall siphon the same way atmospheric pressure pushes the liquid up in a barometer or drinking straw. But of course the energy to lower the pressure at the top is provided by gravity.
Some people think that the atmospheric pressure at the entrance and exit cancel. But actually the atmospheric pressure at the exit doesn't completely cancel that at the entrance because the weight of the liquid in the taller down tube partially cancels some of the atmospheric pressure at the exit. The weight of the liquid in the up tube also cancels some of the atmospheric pressure from the entrance, but not as much because the up tube is shorter, and therefore there is a failure to cancel and a remaining imbalance of atmospheric pressure between the entrance and exit.
So the gravity and air pressure explanation are BOTH CORRECT most of the time and the liquid tensile strength with gravity explanation is ALSO CORRECT in the rare case of siphons at very low pressure, though ONLY at low pressures.