I don't actually think I would take it for granted that "Islam" is the specific proper name as it was intended for what we now refer to as "Islam" as a distinct practice. It can certainly be read as such. So in one of the more famous verses of the Qur'an you could read verse 5:3, as Pickthall does here, as "This day have I perfected your religion for you and completed My favour unto you, and have chosen for you as religion al-Islam."

Or you could just as easily read it more literally as "and have chosen for you submission [to Me] as [the] religion."

Indeed, the word for religion "deen" is synonymous with Islam in the Qur'an, and those who practice it are simply "believers", both words being more frequently used in the Qur'an than either "Islam" or "Muslims."

What's more, when "Muslim" is used, it isn't necessarily applied simply to those who follow the true religion as revealed to Muhammad, so in verse 22:78 for example you have:

He hath chosen you and hath not laid upon you in religion any hardship; the faith of your father Abraham (is yours). He hath named you Muslims of old time and in this (Scripture), that the messenger may be a witness against you, and that ye may be witnesses against mankind.

Of course, that also fits in with the Islamic belief that all of the previous prophets were, in fact, Muslims but their messages (if they were also messengers who came with a book) were corrupted by their followers.

In any event, all of this also gets into questions of just how distinctive and exclusive Islam might have viewed itself, all of which is explored I think quite well in Fred Donner's Muhammad and the Believers.