There are only two version. One is supported by Shia Muslims and one by Sunni Muslims.
Shia Muslims believe God selected Ali S.A. as Caliph and prophet declared this at Ghadir after his final Hajj in front of 120,000 Muslims.
Sunni Muslims believe Abubakr was first Caliph and Ali S.A. was 4th Caliph.
Why is there no single canonical one?
This is and was mostly for political reasons.
Are different versions supported by different groups?
Yes. by shia and sunni
How Abu Bakr, Umar, Uthman and Ali came to lead the ummah?
Abubakr was selected by a meeting of his friends and his tribe when ALi S.A. was at funeral of prophet and was absent in that meeting. That meeting is known as Saqifah Bani Saedeh meeting and in view of Shia it was the base and start of all corruptions in history of Islam including Karbala tragedy and all killings of Imams of Shia and misleading of Ummah.
Umar was selected without meeting nor voting nor shura. But it is said he was appointed by a will letter of Abubakr. But some sources say such letter not really existed and Umar made this letter and stamped the stamp of Abubakr under it while Abubakr was dead. Uthman was selected by a meeting by voting including few of top leaders including Ali S.A. in that meeting Ali S.A. did not receive enough vote and the reasons for not voting to Ali S.A. in that meeting are considered silly by Shia. after some times Uthman abused power and distributed governmental positions among his own tribe and this made Muslims and sahaba and specially Aisha who before was supporter of Uthman angry and finally a group of Muslims (Ali S.A. and his children were not included) seized home of Uthman and finally killed him. Then the population came to Ali S.A> and asked him to accept Caliphate and although he disliked power he finally accepted which his reasoning is said in his sermons in Nahj ul Balaqah (for example see the 3th sermon, Sehqsheqiyah)
AFAIK, the Shiites agree that they did lead the ummah, but say that only Ali was supposed.
Shia believes Ali S.A. was selected by God but they hijacked the government. Shia does not consider their Caliphate legitimate. Shia only consider it a dark happening in history of Islam and only a struggle for power and not consider it legitimate. According to Shia only Allah can appoint the Caliph.
References and more study:
History of The Caliphs, From the Death of the Messenger (S), to the Decline of the 'Umayyad Dynasty 11-132 AH