It's a bit more complex than that! The font made by Monotype for The Times in 1931 was always called Times New Roman - it was, after all, a new Roman-style typeface for The Times.

Linotype then made their own variant simply called Times Roman, which differed mostly in having slanted serifs. The Times switched to this in 1982.

Both Monotype and Linotype produced digital versions, but Linotype's was initially slightly cheaper and thus more popular. In 1984, Adobe licensed it for inclusion in the core Postscript font set, and for a while became the "default" proportional serif typeface.

As WYSIWYG word processing and DTP took off, lots of knock-offs appeared, often called something like "Thymes", "London", or just plain "Roman". At some point, Monotype reduced their prices but by the late 80s Times New Roman was sadly neglected, seen almost as just another clone (after all, The Times itself was using the Linotype version by then!).

TNR became one of the core fonts in Truetype (initially an Apple/Microsoft collaboration, intended to break Adobe's stranglehold on computer typography), so was included by default in Windows 3.1 and Apple's System 7, leading to a resurgence in its popularity.

The Times then moved to Monotype's Times Modern (which features serifs that are even more slanted than Linotype's Times Roman) when they moved to tabloid format in the early 2000s. Monotype bought Linotype at around the same time, so all three fonts are now available from the same source at the same price.