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Lincoln's Inn and Gray's Inn are in the London Borough of Camden, not the City of London. Also they are no longer independent enclaves like the Inner and Middle Temples still are. But yes, they should be (and are) listed seperately, for their special status historically and today (to some extent). David.
Also, may I add, saying that the Inner and Middle Temples "are enclaves not part of the City of London" is not accurate. They are part of the City of London, just like how Lincoln's Inn and Gray's Inn are part of Camden, it's just they are not governed by the Corporation of London. David.