Salamun Alaikum.
I think by saying "Anyone who believes in One God" is a Muslim is an
oversimplification and with respect, incorrect. Some people believe that a man called Jesus is God but this is not the correct concept of God.
The Quran not only demands you believe in one God, but also the correct 'concept' of God with His correct attributes in His
true nature i.e.
One - Alone in his authority and majesty - All knowing, Incomparable, All-Merciful, All-Wise, neither begets nor is begotten, All-Powerful, Master of the Day of Judgment etc.
It was this concept of God that the Quran instilled for the Mushrikeen and its other primary audience. It was not as if the Mushrikeen did not believe in Allah, of course they did. But they thought of Allah as part of a multiple God system (Al-Lat, Al-Uzza, Manat 53:19-20) with Allah as a
Supreme God. It was this concept that the Quran
shattered by asserting that Allah was
Alone and Only
one in His majesty, with
no intercessors or partners.
029.061"If indeed you ask them who has created the Heavens and the Earth and subjected the Sun and the Moon (to his Law), they will certainly reply, "Allah". How are they then deluded away (from the truth)"?039.038“If indeed you ask them who is it that created the Heavens and the Earth, they would be sure to say, "Allah". Say: "Then do you see what you invoke besides Allah? Can they, if Allah wills some penalty for me, remove His penalty? Or if He wills some grace for me, can they keep back his grace?" Say: "Sufficient is Allah for me! In Him do the trusting put their trust"”Similarly, the Quran 'cleaned up' the system / concepts that the Bedouin Arabs followed and brought them in line with the correct 'system' with its underlying principles. Islam as a collection of correct principles and beliefs has always existed and has always been present for man to follow with God's guidance.
On the other hand, by simply saying someone who does not accept the Quran is on the wrong path is also an extreme interpolation. There are many that claim to follow the Quran but are just as much idolatrous as those they rebuke.
So there are many paths to God, such as those that follow their scriptures
in truth. He has given all an open way to race for good. Ultimately the common truth is submission and worship to the
One true God while remaining fully cognizant of His
true nature.
005.048 (part)"...To each among you have we prescribed a law (Arabic: Shir-atan) and an open way (Arabic: waminhajan) . If God had so willed, He would have made you a single people, but (His plan is) to test you in what He has given you: so strive as in a race in all virtues. The goal of you all is to God; it is He that will show you the truth of the matters in which ye dispute”The Quran speaks to mankind as the final testament of God. It lays out correct principles and provides ultimate guidance to follow the
correct 'deen'.
The following article should address your specific question in more detail, God willing. There is a section dedicated to the very question you ask. Please do take a look.
WHAT IS THE TRUE DEFINITION OF 'DEEN' FROM A QURAN'S PERSPECTIVE?http://quransmessage.com/articles/what%20is%20the%20true%20defintion%20of%20deen%20FM3.htmI hope that helps, God willing,
Joseph.