pH is indeed potential of Hydrogen. A lot of the first experiments where the nature of pH was determined involved electrochemistry, and the terminology just kind of stuck even though it turned out not to be electrical and the equations aren't exactly the same. There is a little similarity in that an electron e- plus a hydrogen ion H+ combine to form a neutral atom -- so when electrochemistry was being worked on (which involves the flow and use of e-) they just thought that the way H+ acted would be similar. I don't think that they had realized how buffers work, and the fact that water disassociates.
As posted above, it is probably just best thought of as a function. Just like if we wrote f(x) = cos(x), where now f() stands for "take the cosine of whatever is inside the ()'s", the p in pH stands for "take the negative of the base 10 logarithm of the concentration (in moles per Liter) of whatever comes after the p". Also, it is properly written pH -- little p, big H, not big P big H.
Don't confuse the notion of a partial pressure, which is used primarily in gas mixtures, and the p used in liquid mixtures. They use the same symbol, but are very different concepts. The p in pH has nothing to do with how much hydrogen gas dissolves in water. The partial pressure comes into play when doing gaseous mixtures in equilibrium because the partial pressure of a substance in the gas can almost always be taken as the activity of that substance in the gas. In liquids, the activity is of course a strong function of the mole fraction of the item, but the activity coefficient is usually also a strong function of temperature, pressure, and how the constituents of the liquid behave. And, in ionic mixtures, how many ions are in the mixture can play a role, too, often called the "ionic strength". The only way the partial pressure may play a role in liquid systems is if you are calculating a vapor-liquid equilibrium. Then, maybe both the partial pressure and the pH of the liquid could be important!