You may also like to look at here which says:
Muslim has a general meaning (submission to Allah) and a specific meaning (commonly used now by people around the globe).
The specific meaning still has a large scope as it starts from apparent submission to Allah with minimal state of confession in uniquness of Allah and that the holiness Muhammad --peace be upon him-- in His prophet, to the maximal state of being completely and thoroughly submitted to Allah like the holy prophet was:
قُلْ إِنَّ صَلَاتِي وَنُسُكِي وَمَحْيَايَ وَمَمَاتِي لِلَّهِ رَبِّ
الْعَالَمِينَ
Say: "Truly, my prayer and my service of sacrifice, my life and my
death, are (all) for Allah, the Cherisher of the Worlds [6:162]
But being Mu'min is having faith in heart, it's a good state of being Muslim, both in its general and specific meaning of Muslim, so we both have a believer Christian and a believer Muslim. This attribute further measures the degree someone is Muslim (submitted to Allah, in general or in Islam the last version of the heavenly religion):
قَالَتِ الْأَعْرَابُ آمَنَّا قُل لَّمْ تُؤْمِنُوا وَلَٰكِن قُولُوا
أَسْلَمْنَا وَلَمَّا يَدْخُلِ الْإِيمَانُ فِي قُلُوبِكُمْ وَإِن
تُطِيعُوا اللَّهَ وَرَسُولَهُ لَا يَلِتْكُم مِّنْ أَعْمَالِكُمْ
شَيْئًا إِنَّ اللَّهَ غَفُورٌ رَّحِيمٌ
The desert Arabs say, "We believe." Say, "Ye have no faith; but ye
(only) say, 'We have submitted our wills to Allah,' For not yet has
Faith entered your hearts. But if ye obey Allah and His Messenger, He
will not belittle aught of your deeds: for Allah is Oft-Forgiving,
Most Merciful." [49:14]
although it rather seems that the Christians and other people of book hardly can be considered as Mu'min, most of them can at most be Muslims and at the beginning of being submitted to God, although doing hard worships as well. That's because what they believe in God and His prophet is close to Kufr and Shirk. And if they remedy such ideas then they will commonly convert to Islaam and if they don't live much after remedying their beliefs to convert and die they might be still considered as Mu'min (even although not converted according to the rules of Islam). I rather prefer to skip this issue as I have very small evidences on this issue, and Allah knows best. But let bring these verses that may help:
يَا أَهْلَ الْكِتَابِ لَا تَغْلُوا فِي دِينِكُمْ وَلَا تَقُولُوا عَلَى
اللَّهِ إِلَّا الْحَقَّ إِنَّمَا الْمَسِيحُ عِيسَى ابْنُ مَرْيَمَ
رَسُولُ اللَّهِ وَكَلِمَتُهُ أَلْقَاهَا إِلَىٰ مَرْيَمَ وَرُوحٌ
مِّنْهُ فَآمِنُوا بِاللَّهِ وَرُسُلِهِ وَلَا تَقُولُوا ثَلَاثَةٌ
انتَهُوا خَيْرًا لَّكُمْ إِنَّمَا اللَّهُ إِلَٰهٌ وَاحِدٌ سُبْحَانَهُ
أَن يَكُونَ لَهُ وَلَدٌ لَّهُ مَا فِي السَّمَاوَاتِ وَمَا فِي
الْأَرْضِ وَكَفَىٰ بِاللَّهِ وَكِيلًا
O People of the Book! Commit no excesses in your religion: Nor say of
Allah aught but the truth. Christ Jesus the son of Mary was (no more
than) a messenger of Allah, and His Word, which He bestowed on Mary,
and a spirit proceeding from Him: so believe in Allah and His
messengers. Say not "Trinity": desist: it will be better for you:
for Allah is one Allah: Glory be to Him: (far exalted is He) above
having a son. To Him belong all things in the heavens and on earth.
And enough is Allah as a Disposer of affairs. [4:171]
وَإِنَّ مِنْ أَهْلِ الْكِتَابِ لَمَن يُؤْمِنُ بِاللَّهِ وَمَا
أُنزِلَ إِلَيْكُمْ وَمَا أُنزِلَ إِلَيْهِمْ خَاشِعِينَ لِلَّهِ لَا
يَشْتَرُونَ بِآيَاتِ اللَّهِ ثَمَنًا قَلِيلًا أُولَٰئِكَ لَهُمْ
أَجْرُهُمْ عِندَ رَبِّهِمْ إِنَّ اللَّهَ سَرِيعُ الْحِسَابِ
And there are, certainly, among the People of the Book, those who
believe in Allah, in the revelation to you, and in the revelation to them, bowing in humility to Allah: They will not sell the Signs of
Allah for a miserable gain! For them is a reward with their Lord, and
Allah is swift in account. [3:199]
لَيْسُوا سَوَاءً مِّنْ أَهْلِ الْكِتَابِ أُمَّةٌ قَائِمَةٌ يَتْلُونَ
آيَاتِ اللَّهِ آنَاءَ اللَّيْلِ وَهُمْ يَسْجُدُونَ
Not all of them are alike: Of the People of the Book are a portion
that stand (For the right): They rehearse the Signs of Allah all night
long, and they prostrate themselves in adoration. [3:113]
وَمِنْ أَهْلِ الْكِتَابِ مَنْ إِن تَأْمَنْهُ بِقِنطَارٍ يُؤَدِّهِ
إِلَيْكَ وَمِنْهُم مَّنْ إِن تَأْمَنْهُ بِدِينَارٍ لَّا يُؤَدِّهِ
إِلَيْكَ إِلَّا مَا دُمْتَ عَلَيْهِ قَائِمًا ذَٰلِكَ بِأَنَّهُمْ
قَالُوا لَيْسَ عَلَيْنَا فِي الْأُمِّيِّينَ سَبِيلٌ وَيَقُولُونَ عَلَى
اللَّهِ الْكَذِبَ وَهُمْ يَعْلَمُونَ
Among the People of the Book are some who, if entrusted with a hoard
of gold, will (readily) pay it back; others, who, if entrusted with a
single silver coin, will not repay it unless thou constantly stoodest
demanding, because, they say, "there is no call on us (to keep faith)
with these ignorant (Pagans)." but they tell a lie against Allah, and
(well) they know it. [3:75] [and honesty, truthfulness and
trusteeship are among the most important measures of Iman according to
Ahadeeth, even more important measures than prayer and fast and etc.]
And at a last point, Iman itself has several ranks and it will be measured by Taqwa if I'm not wrong:
يَا أَيُّهَا الَّذِينَ آمَنُوا آمِنُوا بِاللَّهِ وَرَسُولِهِ وَالْكِتَابِ الَّذِي نَزَّلَ عَلَىٰ رَسُولِهِ وَالْكِتَابِ الَّذِي
أَنزَلَ مِن قَبْلُ وَمَن يَكْفُرْ بِاللَّهِ وَمَلَائِكَتِهِ وَكُتُبِهِ
وَرُسُلِهِ وَالْيَوْمِ الْآخِرِ فَقَدْ ضَلَّ ضَلَالًا بَعِيدًا
O ye who believe! Believe in Allah and His Messenger, and the scripture which He hath sent to His Messenger and the scripture which
He sent to those before (him). Any who denieth Allah, His angels, His
Books, His Messengers, and the Day of Judgment, hath gone far, far
astray. [4:136]
يَا أَيُّهَا الَّذِينَ آمَنُوا اتَّقُوا اللَّهَ وَآمِنُوا بِرَسُولِهِ يُؤْتِكُمْ كِفْلَيْنِ مِن رَّحْمَتِهِ وَيَجْعَل لَّكُمْ
نُورًا تَمْشُونَ بِهِ وَيَغْفِرْ لَكُمْ وَاللَّهُ غَفُورٌ رَّحِيمٌ
O ye that believe! Fear Allah, and believe in His Messenger, and He will bestow on you a double portion of His Mercy: He will provide
for you a Light by which ye shall walk (straight in your path), and He
will forgive you (your past): for Allah is Oft-Forgiving, Most
Merciful.
There are many verses that Allah tells the believers to believe, to fear Allah, and warn them, but there is no where that Allah warns a Mutaqqi who fears Allah as this fear is itself a mean to save the Mu'min.
PS. there may exists gaps in this answer, so feel free to fill them. Also bear in mind that the writer of the answer is not a scholar/expert and can easily make mistakes, although tries to prevent errors.
