1.Islam and Muslim are both words used to describe the religion revealed to the Prophet Mohammed.In other words, according to your source, while there are differences in usage of the two words, it is quite fair to say that "all Muslims are Islamic", as a shorthand for "all Muslims follow the Islamic religion".2.Islam and Muslim both have the same origin in the Arabic verb s-l-m.
3.Islam is the act of submitting to the will of God whereas a Muslim is person who participates in the act of submission.
4.To be correctly used, Islam or Islamic should describe the religion and its subsequent cultural concepts whereas Muslim should only describe the followers of the religion of Islam.
If you're just arguing syntax, that's fine, and I agree with you. A person is a Muslim, the religion is Islam. But semantically, any Muslim is a practitioner of Islam, by the definition of the words involved.
Just surrender and be done with it.