Hi Kyle!
I have found the problem and it might occur for other developer as well, so I want to share my experience:
In the app settings, I had inserted my web site domain in the field "App domain" ("basic" tab).
That was obviously a mistake.
I took that entry away and instead I wrote my web site domain into the field "Valid OAuth redirect URIs" ("Advanced" tab). Until today, I had another value in this field: that value standing in the CB plugin parameter, in the field "Callback URL", which I for some reason understood to be the value needed.
After this change, the login works perfectly from any device and from any browser.