Hello,
i had the same problem some time ago too. The shelly_cloud_api_access.pdf is not really helpful at that point. I don't know if there is a better documentation out there but i was able to solve the problem:
The key point is that you have to make a https POST request and not a GET request to switch the plug. You can easily try it out using wget:
wget -q -O - --post-data "id=xyz&auth_key=abcxyz&channel=0&turn=on" https://shelly-11-eu.shelly.cloud/device/relay/control
If you specify your device id and auth_key, it should work.
I am controlling my Shelly Plug S using the python requests library. If you like to have some examples on that just write me.
Nevetheless a GET request is working also, if you only want to get the status information. So this is what makes it really confusing, that it works for one thing but does not work for the other things.