In 2019 I had written down 1 connector listed on Tesla website. Nowadays it lists 2 connectors, up to 22kW.
Chargemap doesn't show any picture. On GoingElectric there are pictures, likely from sometime between July and September 2018 showing 2 Tesla chargers, one with the red sign and one with the grey all-EV sign. There are no marked parking spots, it's some asphalt around a hangar.
I can't find any information about parking or charging on the website https://www.gstaad-airport.ch/