I think your question is perfectly fine, I don't think this forum is only for advanced research level questions.
Assuming you are not actually receiving small droplets of water, the air around the droplets is cooled by the water, which will then cool your skin. This is strongly accentuated by the fact the water create air currents.
While the effect you suggest of feeling an absence of radiating heat is physically sound, my intuition would tell me it is negligible in this case. Radiation of objects at room temperature wouldn't be very strong compared to the diffusive and convection effects induced by your cold shower.
Short answer:
The thermometer measures actual temperature (which is the same for both), while your hand measures the transfer of energy (heat), which is higher for the pot than the air.
Long answer:
Keyword: Thermal Conductivity
The difference is a material-specific parameter called thermal conductivity. If you are in contact with some material (gas, liquid, solid), heat, which is a form of energy, will flow from the medium with higher temperature to the one with low temperature. The rate at which this happens is determined by a parameter called thermal conductivity. Metals are typically good heat conductors, which is why metal appears colder than air, even though the temperature is the same.
Regarding your second question: the thermometer will show the same temperature. The only difference is the time at which thermal equilibrium is achieved, i.e. when the thermometer shows the correct temperature.
Final remark: the rate at which heat (energy) is drained from your body determines whether you perceive a material as cold or not, even if the temperature is the same.
For reference, here is a table which lists thermal conductivities for several materials:
Best Answer
Yes, the deodorant contains a mixture of low boiling point alkanes, such as butane, that form a liquid under pressure but evaporate when the pressure falls to one atmosphere as they leave the can. It's primarily the latent heat of vaporisation that reduces the temperature and makes it feel cold.
In addition to this, deodorants (as opposed to antiperspirants) contain ethanol - they are essentially just a solution of perfume in ethanol plus propellant. The ethanol evaporates on the skin and again the latent heat of vaporisation cools the skin.
In a can of deodorant the dip tube goes down into the liquid propellant. When you press the button the pressure in the can forces liquid alkane up the dip tube and out. The alkane mostly evaporates in the tube and nozzle, but if you hold the can very close to your skin you can get liquid alkane on the skin. This evaporates very rapidly, and it's really cold!