Why is an alcohol proton so shielded?
Oh, I was just reading about this. This excerpt points out the factor contributing to the chemical shift observed in alcohols


  The extent of hydrogen bonding varies with concentration, temperature, and solvent, and changes in the degree of hydrogen bonding can cause substantial shift changes. This is very evident in the nmr spectrum of ethanol taken at different concentrations in CC14 (Figure 9-29). The hydroxyl resonance will be seen to move upfield at the lower concentrations. This is the result of decreasing association by hydrogen bonding through equilibria...


Source

So basically the reason (or a reason) for alcohols moving upfield due to hydrogen bonding, which results in decreasing association.

I looked at your link and there are a few notable things. They mention *variable, highlighting that it is not necessarily constant based on different conditions, which connects to hydrogen bonding. For carbon-2, we have the normal ~1.7 for methine protons + 2.5 from oxygen of alcohol, coming to be a pretty close value. So no, it's not "why on earth would the electrons on the carbon-2" but what on earth is going on with that alcohol?