The quoted definition may sound confusing because of the use of "vapour pressure", which is not necessarily related to the liquid-air surface. This can be made clearer by a different example:
Suppose you have a water container at room temperature and you heat only a small volume at the centre of the container (by using focused radiation of some sort, for example). That heated volume would be considered as "boiling" when its phase changes to gas phase (due to the elevated temperature), and the pressure of that gas is higher than the surrounding pressure (hydrostatic + atmospheric).
In this example, "vapour pressure" is the (elevated) pressure of the heated volume, and "external pressure" is the sum of hydrostatic and atmospheric pressures.
Also, as @brandon-enright mentioned,
That's why you don't see boiling even around thermal vents in the
bottom of the ocean. The water just superheats.
So boiling is not a surface phenomena, as it can happen far away from any interface and quench far before reaching any interface.