Confusion about boiling point and vapor pressure
Shouldn’t we say that the liquid boils when its vapor pressure is equal to the greatest fluid pressure in the liquid?


No.  If you heat up water at the ocean's surface to 100 &deg;C, it will boil, even though the pressure at the bottom of the ocean is far higher.

As you note, pressure is not guaranteed to be homogeneous in space (i.e. the same everywhere), especially for deep bodies of water.  But neither is nearly any other fluid property.  Temperature can vary with position in poorly-mixed waters.  So can whether the water is boiling or not.  

A given small-ish region of water with nearly constant temperature and pressure will boil if and only if the (P, T) condition of the water is at or above the water's vapor pressure at the same temperature.