In the Quran, there's a clear separation between believers and disbelievers, but it doesn't talk of doubters as widely to best of my knowledge. Given that, I wonder whether there are references to doubters either in the Quran or hadiths, and whether Islam differentiates between doubters and disbelievers in any way. By doubter, I mean someone who doubts the correctness of Islam but does not necessarily thinks that it's false or/and working against it.
Doubt (شك) is classed as a type of disbelief. Various scholars have cited consensus on that.
واتفقوا ... فان من جحد شيئا مما ذكرنا أو شك في شيء منه ومات على ذلك فانه كافر مشرك مخلد في النار أبدا
And they are agreed ... he who denies anything of what has been mentioned or doubts in anything of it and then dies in that state, then he is a disbeliever and polytheist who shall remain in the fire forever
There are verses in the Quran which have counted doubt as the trait of the disbelievers and those that have have contrasted it with belief:
لنعلم من يؤمن بالآخرة ممن هو منها في شك
That We might make evident who believes in the Hereafter from who is thereof in doubt.
إنما المؤمنون الذين آمنوا بالله ورسوله ثم لم يرتابوا وجاهدوا بأموالهم وأنفسهم في سبيل الله
The believers are only the ones who have believed in Allah and His Messenger and then doubt not but strive with their properties and their lives in the cause of Allah.
Does Islam differentiate between such an individual, and someone who denies Islam due to his caprices, or someone who does not even bother studying Islam?
Islam may differentiate but it is God the Most High, the Most Merciful Who will decide:
And if We give him a taste of prosperity, after some adversity has afflicted him, he will say, “Troubles have gone away from me.” He becomes excited and proud. Except those who are patient and do good deeds—these will have forgiveness and a great reward. Perhaps you wish to disregard some of what is revealed to you, and you may be stressed because of it, since they say, “If only a treasure was sent down to him, or an angel came with him.” You are only a warner, and God is Responsible for all things.
The final question goes beyond our competences, as it is God Who has the final answer. I can only answer from what I have understood from the Word of Him. I only cite the Word of God according tho the Holy Quran.
God promises Paradise to those who do believe and do good:
As for those who believe and do good deeds, We will admit them into Gardens beneath which rivers flow, abiding therein forever. They will have purified spouses therein, and We will admit them into a shady shade.
God warns of punishment to those who know but cease to believe
O you who believe! If you obey a party of those who were given the Scripture, they will turn you, after your belief, into disbelievers. And how could you disbelieve, when God’s revelations are being recited to you, and among you is His Messenger? Whoever cleaves to God has been guided to a straight path. O you who believe! Revere God with due reverence, and do not die except as Muslims. And let there be among you a community calling to virtue, and advocating righteousness, and deterring from evil. These are the successful. And do not be like those who separated and disputed after the clear proofs came to them; for them is a great punishment.
It is not excluded but not promised that those will be pardoned:
Do not apologize. You have disbelieved after your belief. If We pardon some of you, We will punish others, because they are guilty.
Those who do not believe in the Prophet ﷺ but believe in God and do good deeds may expect to be received:
There shall be no compulsion in religion; the right way has become distinct from the wrong way. Whoever renounces evil and believes in God has grasped the most trustworthy handle; which does not break. God is Hearing and Knowing.
They believe in God and the Last Day, and advocate righteousness and forbid evil, and are quick to do good deeds. These are among the righteous.
To God belongs whatever is in the heavens and whatever is on earth. He will repay those who do evil according to their deeds, and recompense those who do good with the best.
Those who doubt but believe in God are between those who reject and those who believe.
As I initially stated, God will be the One to judge. It would be wrong to answer this question.