Real boiling temperature I've been thaught that water at 1 atm water boils at 100 °C (the so called “normal boiling point”). 

However, I came across a more rigorous description of the boiling process saying that in order to fulfill thermodynamic equilibrium between liquid and vapour the pressure inside the bubble must be higher than that outside and accordingly the liquid results superheated (to be at the same temperature of the vapour).

I also understand the nucleation process, but here for sake of simplicity I assume that the bubble is already present.

So my question is why we say that water boils at 100 °C (at 1 atm)?
Simply speaking: The rising bubbles in your pot shrink wherever it is too cold (&lt; 100 °C), and grow whereever it is too hot (> 100 °C), equillibrating the temperature, by exchanging some heat of evaporation. If you keep your thermometer at a distance above the heat source, it will show (mostly, see below) exactly 100 °C, or whatever is the boiling point at the given outside pressure.

Very small bubbles have to have a measurably increased inside pressure, but while the water is boiling, the bubbles grow, and so this difference vanishes. You can calculate it with the surface tension of water (0.06 N/m at 100 °C) using the https://en.wikipedia.org/wiki/Kelvin_equation, and it is negligible for bubbles of millimeter size.

Of course with increasing distance to the water surface, the temperature rises  because of the pressure increase of ~ 0.1 bar/m. Slightly less, because water at 100 °C is less dense.