You mean that we can say that as "different nations" have a common language, so this language become "international"?
And, since "different tribes" have a common language, so this language could be "lingua franca"?
I think you're belabouring your question; the definitions have been given clearly and correctly -- you just seem to want some other answer..... sorry, but I think you just need to read the explanations given once again. International means between a number of countries, all of which use the same language.
German, for example, is used in Germany, Switzerland, Austria and Luxemburg.