The following is a narration from Sahih of Imam Muslim:
Book 5, Number 2333:
Zaid b. Wahb Jahani reported and he was among the squadron which wall under the command of Ali (Allah be pleased with him) and which set out (to curb the activities) of the Khawarij. ‘Ali (Allah be pleased with him) said: 0 people, I heard the Messenger of Allah (may peace be upon him) say: There would arise from my Ummah a people who would recite the Qur’an, and your recital would seem insignificant as compared with their recital, your prayer as compared with their prayer, arid your fast,as compared with their fast. They would recite the Qur’an thinking that it supports them, whereas it is an evidence against them. Their prayer does not get beyond their collar bone; they would swerve through Islam just as the arrow passes through the prey. If the squadron which is to encounter them were to know (what great boon) has been assured to them by their Apostle (may peace be upon him) they would completely rely upon this deed (alone and cease to do other good deeds), and their (that of the Khawarij) distinctive mark is that there would be (among them) a person whose wrist would be without the arm, and the end of his wrist would be fleshy like the nipple of the breast on which there would be white hair. You would be marching towards Muawiya and the people of Syria and you would leave them behind among your children and your property (to do harm). By Allah, I believe that these are the people (against whom you have been commanded to fight and get reward) for they have shed forbidden blood, and raided the animals of the people. So go forth in the name of Allah (to fight against them). Salama b. Kuhail mentioned that Zaid b. Wahb made me alight at every stage, till we crossed a bridge. ‘Abdullah b. Wahb al-Rasibi was at the head of the Khwarij when we encountered them. He (‘Abdullah) said to his army: Throw the spears and draw out your swords from their sheaths, for I fear that they would attack you as they attacked you on the day of Harura. They went back and threw their spears and drew out their swords, and people fought against them with spears and they were killed one after another. Only two persons were killed among the people (among the army led by Hadrat ‘Ali) on that day. ‘Ali (Allah be pleased with him) said: Find out from among them (the dead bodies of the Khwarij) (the maimed). They searched but did not find him. ‘Ali (Allah be pleased with him) then himself stood up and (walked) till he came to the people who had been killed one after another. He (‘Ali) said: Search them to the last, and then (‘Ali’s companions) found him (the dead body of the maimed) near the earth. He (Hadrat ‘Ali) then pronounced Allah-O-Akbar (Allah is the Greatest) and then said, Allah told the Truth and His Messenger (may peace be upon him) conveyed it. Then there stood before him ‘Abida Salmani who said: Commander of the Believers, by Allah, besides Whom there is no god but He, (tell me) whether you heard this hadith from the Messenger of Allah (may peace be upon him). He said: Yes, by Allah, besides Whom there is no god but He. He asked him to take an oath thrice and he took the oath.
As seen from the narration, Ali ibn Abi Talib رضي الله عنه was convinced of them being the foretold Khawarij because they have shed forbidden blood and the raided the wealth of Muslims. The obvious distinction between the Khawarij from other everyday criminals are that the Khawarij justified it in the name of Islam (and this by making false takfir of Muslims) and considered it as their religious duty and right.
Why is this important ? Because one of the excuses made by the modern day Khawarij is to say that “the Khawarij considered those who committed sins to have committed kufr and we do not ascribe kufr for committing sins, hence we are not the Khawarij”. This argument of theirs is false. As seen in the above narration, Ali ibn Abi Talib رضي الله عنه did not investigate whether they ascribed kufr on those who committed sins. In fact, the Khawarij did not emerge at the time of Ali ibn Abi Talib رضي الله عنه based on this doctrine. Instead they ascribed Kufr based on saying that “Judgement belongs to Allah alone” and that Ali Ibn Abi Talib رضي الله عنه committed kufr by judging by other than Allah. And Ibn Umar رضي الله عنه said about the Khawarij that they used to take the verses revealed regarding the polytheists and ascribe it to believers i.e. they take verse revealed in context to polytheists and strip it of the that context and meaning and details and ascribe it to Muslims instead based on simple literal reading . The doctrine of imputing disbelief for committing sins was instead a later day formation when the Khawarij crystallized into a separate isolated sect. This doctrine is not mentioned as the trait of the Khawarij in any of the Prophetic narrations describing them. The scholars only cited it as it was one of the important doctrine of the later Khawarij sects and not on the basis that it is a condition for one to be Khawarij. In fact, the Hanafi jurist, Imam Ibn Abideen (d. 1836) similarly explains when he went he went onto describe the Wahhabi movement to be a Khawarij sect:
“His words and who consider the Companions of our Prophet (Allah bless him and give him peace) to be disbelievers are not a condition for someone to be a kharijite, but rather are a mere clarification of what those who revolted against ‘Ali (Allah Most High be well pleased with him) in fact did. Otherwise, it is enough to be convinced of the unbelief of those they fight against, as happened in our own times with the followers of [Muhammad ibn] ‘Abd al-Wahhab, who came out of the Najd in revolt, and took over the sanctuaries of Mecca and Medina. They followed the Hanbali madhhab, but believed that they were the Muslims, and that those who believed differently than they did were polytheists (mushrikin). On this basis, they held it lawful to kill Sunni Muslims (Ahl al-Sunna) and their religious scholars, until Allah Most High dispelled their forces, and the armies of the Muslims attacked their strongholds and subdued them in 1233 A.H. [1818CE].” (Hashiya Radd al-Muhtar, 4.262)
So it not a condition for one to be Khawarij that they have to consider Ali ibn Abi Talib رضي الله عنه or Uthman bin Affan رضي الله عنه in specific as disbelievers, and similarly not a condition that they have to specifically believe that committing sins are kufr. The Khawarij on the contrary are those who make false takfir of Muslims and on account of it fight the Muslims shedding their blood and raiding their wealth.