By lat/long:
North: Keflavik Airport, Iceland
South: Singapore
East: Narita Airport, Japan
West: Off the western shore of Kauai, HI
By heading:
North: Keflavik Airport, Iceland
South: Singapore
East: Binsar, Uttarakhand, India
West: Lat Krabang, Thailand
If layovers don't count, then Denali National Park instead of Keflavik for North and Hanoi instead of Narita for East by longitude.
I would count layovers, and here's a some reasons to consider.
If you would count travel in a car, the why not in a plane on the ground? In both cases you are separated from the earth by pavement, tires, and the body of a vehicle.
If you changed planes at the airport, why is the terminal different from other buildings? Is there any difference between stepping out of a car in a garage, off a boat at a dock, or a plane on the tarmac?
If we are talking geographic extremes on the surface of the Earth, then why not treat all of the surface equally? If you weren't at those coordinates on the surface at that time, where were you?
I've disagreed with you on this subject before, and I'll set out my reasons here.
I have in the past flown the Dragonair route from Dhaka to Hong Kong with a thirty-minute stop in Kathmandu. I would not say that I have been to Nepal. I have never passed through Nepalese customs and I have never received a Nepalese stamp in my passport. My experience in Nepal was sitting in an airplane and looking out the window and seeing Nepalese land. A few minutes prior, I was sitting in an airplane and looking out the window and seeing Indian land. Neither you and I would say that my flying over India would enable me to count India, but in terms of my actual experience with the country, what's the difference?
My not counting airports is arbitrary, yes, but so is your counting everything on Earth's surface but nothing in Earth's atmosphere. It's just a matter of different points of view.