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Re: "I may be wrong, but I believe the name of the type family is simply Times New; the name of the italic face would then be Times New Italic rather than the contradictory Times New Roman Italic."

The name of the typeface really is Times New Roman. The roman variant is called Times New Roman Regular and the italic, Times New Roman Italic (which I agree is awkward).

The reason for this is trademarks: "Times" was a registered trademark of Linotype and so when Monotype developed a similar typeface, they used and trademarked "Times New Roman."

Some time ago, I forget when, Monotype acquired Linotype and so is now the holder of both trademarks.

Don't know if Butterick's article mentions this or not. I've read it but not recently.

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Replying to my own comment because I'm wrong. I thought that Times came first and Times New Roman later, but according to this article from the New York Public Library, the typeface was designed only once (by Stanley Morison).

The metal punches were then created jointly by Linotype and Monotype, which each sold these under different names (and registered trademarks), Times by Linotype and Times New Roman by Monotype.

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Forgot the link in the previous comment:

https://www.nypl.org/blog/2014/12/09/times-new-roman