Not just atmospheric pressure
Pressure isn't just caused by the atmosphere; it's also caused by the column of liquid. I do not say that gravity doesn't cause the siphon effect, but that the siphon effect could not happen in a vacuum, as the pressure of the liquid alone would not be enough to move the liquid over the 'hump'. A vacuum bubble would appear at the same level in the tube as at the surface of the reservoir.
The only reason that the liquid and atmosphere cause pressure at all is gravity. Gravity also makes liquid flow downhill; this is not a factor of pressure but rather the transfer of gravitational potential to kinetic energy. When the liquid is flowing uphill – as in the part of the siphon before the 'hump' – its gravitational potential is increasing: gravity is working against the flow of the liquid in this section.
The force driving the liquid up into the 'hump' is a static pressure force exerted on the liquid in the bottom of the tube by the sum of the liquid pressure and the atmospheric pressure above it. This force encourages the liquid up the tube. Because of gravity, the weight of the liquid within the first stage of the tube is also exerting a downward pressure force; this force works against the other force. The difference in force which enables the liquid to flow over the hump is caused by atmospheric pressure.
The liquid is denser than the atmosphere, so exerts a greater static pressure for a given volume (or height, if you will). Once the liquid is over the hump, it does indeed fall due to gravity, rather than the atmospheric pressure. But the flow is maintained because the static pressure at the bottom of the siphon is higher than the static pressure in the receptacle; while the atmospheric pressure may be slightly higher at the lower altitude, the liquid pressure is higher still, as the liquid is denser than air.
The problem is that, in the absence of any atmospheric pressure, the liquid simply cannot get over the hump. Without atmospheric pressure, there is no force to drive the liquid against the force of gravity. And if you are thinking 'suction'... well, suction is pressure difference, and there can never be less pressure than zero, so it wouldn't exist in a vacuum.
(*Aside* There are other things going on too, of course; I notice that some have cited surface tension, by which I assume they mean the bonding between the liquid molecules; surface tension only happens at a surface, and there isn't one inside a siphon. But liquid bonding forces are weak compared with pressure forces; that's what makes liquids liquids. Dynamic pressure, too, undoubtedly plays a part, particularly in the downward portion of the tube, but I'll leave discussion of that to fluid dynamicists.)
So the siphon – that part of the mechanism that gets the liquid over the edge of the reservoir, rather than the part which delivers it into the bucket – is, arguably, driven by atmospheric pressure.
I accept that you need gravity for a siphon to work. (Though I should point out the obvious here – the whole concept of a siphon is meaningless in zero gravity: which way is up?) But I will strenuously argue that it won't work in a vacuum, which means that atmospheric pressure is an essential element in the process.
In short, asserting that a siphon isn't driven by atmospheric pressure is more wrong than asserting that it is.