Izanagi is a technique available to all Uchiha members with the Sharingan, but it has various levels of effectiveness:
The Uchiha, descended from the Sage, are able to perform Izanagi with their Sharingan ... To use Izanagi to its full potential, users must also have the genetic traits of the Senju, who are also descended from the Sage.
So, Itachi didn't lie, Obito didn't lie, and it is not a plothole. You can see how my answer clashes with the accepted answer.
For normal Sharingan users, Izanagi can help you change reality for a brief moment.
Incomplete Izanagi (it's better)
For Sharingan users with Hashirama's DNA, it extends the lifespan of the jutsu. Since Danzo had Senju DNA in his right arm, he was able to extend Izanagi for each eye up to a minute.
Danzō had some of Hashirama Senju's DNA transplanted into his arm, which extended the time span of each Sharingan's Izanagi to a minute. However, because Danzō is not an Uchiha, his chakra levels drop substantially every time a new Sharingan is used to activate the technique.
The Complete Form of Izanagi (the best)
Finally, we have the Uchiha members who can control Hashirama's DNA — Obito and Madara. Obito's Izanagi can last much longer than Danzo's (surviving Konan's signature move), and he doesn't need to use any hand seals. As for Madara, he never uses Izanagi with Senju's DNA transplanted into him, although it is shown that Senju's DNA heals his right eye in the end.
Obito also makes use of Hashirama's DNA but, in contrast to Danzō, he claims to have gained control over Hashirama's DNA and as such, boasts the completed form of Izanagi. Showing evidence to this claim, Obito could maintain this technique's power much longer with a single eye than Danzō and could perform it with no hand seals at all.
Madara's Izanagi Trick
Madara had the Izanagi activated after his death, and his blind eye was healed by Hashirama's DNA.
But Madara had planned ahead: he had scheduled an Izanagi to activate sometime after his death, changing reality to bring him back to life in exchange for his right eye's vision ... went into hiding with ... a mouthful of Hashirama's flesh that he transplanted into his wounds .. in the process restoring his right eye.