You expect that the paper will bend downwards due to the decreased pressure applied to the air gap causing an increase in its volume, but your observation is that instead the paper is bent upwards.
I think that this is probably caused by water leakage. Try the experiment again with a plastic seal over the glass, I expect you to see the film bend down. As an estimate of how much down, air pressure is around 30 feet of water. Your glass is holding a few inches. So I would think this fraction (2/12)/30 = 1/180 would be about the amount of movement, on average. For 2 inches of water this would be about 1/90 of an inch, which is detectable (look to see how reflections in the film are distorted).
Once again, being at a restaurant with friends, I go to drink iced-water from a drinking glass again, and sure enough, this very thing happened again. Perturbed and determined to figure why ice fuses together like this, the experimentalists came out in me and I started pouring water into glasses full of ice when I got home. I believe that I have an answer to my own question.
Even though an ice water mixture maybe at 0℃, the temperature of the ice depends on how long the ice has been in the glass of water. If the ice has been in the water for some time, then the ice eventually will reach it maximum temperature of 0℃. However, ice fresh out the ice freezer, will be colder (FDA recommends -20℃) in the inside than the outside and this holds the key to why ice will fuse together in a glass of water.
Fresh Ice at -20℃
Starting off with ice at -20℃ and pouring water into the drinking glass, the edges of the ice and water will be in thermal equilibrium at 0℃ after a short period of time. However, in order to freeze the water in-between the ice cubes, latent heat must be removed in order to freeze this very water. (Remember that latent heat of fusion is the required amount of heat that must be removed to change water into ice.) This is an image I created to show you what I am thinking here.
The dark blue regions indicate the colder -20℃ internal regions of the ice cubes whereas the light blue regions indicate the warmer 0℃ regions of the water as well as the edges of the ice cubes. The red arrows indicate the directions of the latent heat of fusion being removed from the water, and therefore, freezing the water in-between the ice cubes. I also see no reason why ice will not fuse to the sides of the glasses as well, using this same mechanism to remove the required latent heat. To test this, I poured water into glasses filled with fresh ice into several drinking glasses filled with water and was able to get the ice to fuse.
“Old Ice at 0℃”
Starting with fresh ice again, I now constantly stirred the ice around so that I did not allow them to fuse together, and after about 10 minutes or so, I let the ice sit in the glass undisturbed. I found that the ice never fused together. This makes sense because I think that the ice is now at 0℃ throughout the whole ice cube. As a result, there is no way to remove the required latent heat to freeze the water in-between the ice cubes or the sides of the drinking glass.
Best Answer
What happens is that the water forms a seal between the steel glass and the glass table.
Due to the slightly higher pressure $p$ inside the glass it's kept afloat while the inside pressure is maintained because of the seal. The slight over-pressure arises when the glass 'sinks' into the water layer and is caused by the weight of the glass, which slightly compresses the air 'trapped' between the glass and the seal. The over-pressure ensures the glass never really touches the table, making movement almost frictionless.
Water also has quite a low viscosity compared to many other liquids, so with the seal in place, the steel glass is quite mobile with respect to the glass table.
But that is all these is to this phenomenon.