A non-believer and pagan may not be the same in Islam. It is better to use the Arabic word.
In Islam, there are some similar words:
Ahl Kitab (people of Book): they are Christians and Jews who have holy Book from God.
Mushrik (polytheists): they are who worship anything unless Allah or associate anything with God and consider divine power for anything unless Allah. for example idol worshipers.
Furthermore, according to Islam: in original Christianity, there is no Trinity and it was added later as a deviation. In Islam believing to Trinity is polytheism.
The Qur'an says:
[4:171] Sahih International
O People of the Scripture, do not commit excess in your religion or
say about Allah except the truth. The Messiah, Jesus, the son of Mary,
was but a messenger of Allah and His word which He directed to Mary
and a soul [created at a command] from Him. So believe in Allah and
His messengers. And do not say, "Three"; desist - it is better for
you. Indeed, Allah is but one God. Exalted is He above having a son.
To Him belongs whatever is in the heavens and whatever is on the
earth. And sufficient is Allah as Disposer of affairs.
[5:73] Sahih International
They have certainly disbelieved who say, "Allah is the third of
three." And there is no god except one God. And if they do not desist
from what they are saying, there will surely afflict the disbelievers
among them a painful punishment.
Kafir: literally means farmer because covers on the see. It is who know truth does not accept it. in fact covers on it. it can be people of Book or polytheists. In fact, a Kafir is any kind of non-Muslim.
In Islam, a real Christian if know true Islam without propaganda accepts Islam and converts Islam. and has the duty to find truth.
In Islam, any human has the duty to do enough research to find truth and such human will find Islam as truth. but who did enough try and research but still dic not find truth is forgiven by God.
The Qur'an says:
إِنَّ الَّذِينَ تَوَفَّاهُمُ الْمَلَائِكَةُ ظَالِمِي أَنفُسِهِمْ قَالُوا فِيمَ كُنتُمْ ۖ قَالُوا كُنَّا مُسْتَضْعَفِينَ فِي الْأَرْضِ ۚ قَالُوا أَلَمْ تَكُنْ أَرْضُ اللَّـهِ وَاسِعَةً فَتُهَاجِرُوا فِيهَا ۚ فَأُولَـٰئِكَ مَأْوَاهُمْ جَهَنَّمُ ۖ وَسَاءَتْ مَصِيرًا
[4:97] Sahih International
Indeed, those whom the angels take [in death] while wronging themselves - [the angels] will say, "In what [condition] were you?" They will say, "We were oppressed in the land." The angels will say, "Was not the earth of Allah spacious [enough] for you to emigrate therein?" For those, their refuge is Hell - and evil it is as a destination.
Believers is who believes in unseen and can be said about Christians. in fact we should ask believer to what? a Muslim is believer to Allah and unseen things described in Quran.
One can believe in 20 gods but is not Muslim.
Whoever follows his Aql (reasoning and logic and wisdom) will go to heaven.
Aql says do not lie, do not steal, be honest, be good, increase your knowledge,... Aql has 80 element in its army and Jahl (Ignorance) has 80 element in his army and they are always fighting each other in mind of human.
And Aql says Islam is truth. So then who follow instructions of Islam go Heaven.
'Aql is internal messenger of God in each human.