How is the temperature in which vapor pressure is equal to atmospheric pressure the boiling point of the liquid?
Vapour pressure is not merely calculation, it is easily measureable.
Evaporation is due molecules leaving liquid not by overcoming atmospheric pressure, but by overcoming cohesive intermolecular forces. The saturated vapour pressure is a measure of the rate of this massive leaving of molecules.
Boiling is just a special evaporation case. Above boiling point, vapour is always unsaturated and liquid evaporate at its surface regardless of gas composition. Additionally, random processes create miniature bubbles ( or these may be bubbles of formerly dissolved gasses ), that have the same or higher pressure then external pressure and expands themselves.
The result is more or less massive and turbulent evaporation with creation of vapour bubbles.
Liquids can be frequently overheated above their boiling point, if there are absent &quot;bubble seeds&quot;. Once available, e.g. by strirring or repulsion of dissolved gas, creation of vapour bubbles quickly cools liquid down to its boiling point.