I don't understand what the wiki page means by "internal compression". That sounds like nonsense. FWIW this is also an issue in tuning pipe organs, and in the early 20th century reputable organ-builders were writing books stating that the speed of sound was faster in a narrow bore pipe than a wide bore one, which doesn't make any sense - but these bits of mythology linger on.
The formula L = (S/F) / 4 makes the assumption that the pressure at the open end of the tube is always exactly atmospheric. That is only an approximation, because the air outside the end of the tube also vibrates as well as the air inside. (If the air outside did not vibrate, you wouldn't hear any sound from the instrument).
For a reasonably "thin" pipe, you can get a better approximation to the pitch by taking the effective length of the pipe as slightly longer than the physical pipe, to account for the vibration of the air outside. For a cylindrical pipe a long distance from anything that interferes with the air flow, the correction factor for the length is about 0.3 times the diameter.
The airflow at the end of the pan pipes will be affected by the the player's face blocking part of the air flow and the player blowing across the end of the pipe, so presumably 0.82 times the diameter is a better correction factor than 0.3 for this situation.