Why Local Noon?
The answer to this question is related to the answer to why do the two sites need to be on the same longitudinal line. The answer to this latter question is both yes and no.
The sites do have to be on the same longitudinal line, but they don't have to be there at the same time. For example, in New York today (July 15), the sun rose at 5:38 am and will set at 8:26 pm. This means that there were 14 hours and 48 minutes of daylight. Local noon is the midpoint of that daylight span or 1:02 pm (DST).
Puerto Rico is about 8 degrees east of New York. Since the earth rotates
west to east, Puerto Rico will experience "local noon" first. Since it takes
the earth about an hour to travel 15 degrees it will take approximately 30
minutes for New York to move into position to experience local noon.
Therefore, local noon in Puerto Rico today is about 12:30 assuming Daylight
Savings Time as well since both cities are in the same time zone.
In other words, local noon is not a particular configuration of the hands on a clock, but rather a physical event. Local Noon when the sun is highest in the sky.