The T-62 is the forgotten Soviet Tank. There doesn't seem to be any information about its variants and there doesn't seem to be any agreement on them when there is information.
Any information on T-62 history and variants?
Soviet/Russian Armor and Artillery Design Practices - 1945-1995, while perhaps not definative, has as good a right up on the history of T-62 as anything I've got.
According to this, the Vagonka design bureau proposed to further develop the T-55 because that model did not improve on the T-54 in either firepower or armor. Work started about 1957 - several years before the M-60 became a threat. Originally, the T-62 was planned to have a more powerful D-54TS 100mm gun with longer tube and more powerful ammo. There was a competing 100mm smoothbore gun which Kruschev favored and insisted be adapted to tanks. It was not well suited for tanks so the ammunition was shortened and the caliber changed to 115mm to become the 2A20. T-62 was basically ready for production in 1960 but had stabilizer problems and the project was mired in political controversy since it competed with Kharkov's more revolutionary (but 100mm gun armed) Obiekt 430 design, all of which cause delays.
When the M-60 was issued to U.S. Army Europe in December 1960 and the Soviet's completed their technical evaluation a month later they were peeved to learn the M-60 had a 105mm gun - bigger than their tanks 100mm gun and the frontal armor could not be penetrated by the 100mm. This finally got things moving on the T-62 and production started in the later half of 1961. So while the T-62's design was not a reaction to the M-60 (it was what in the west would be called a "private-venture" built without a military requirement), it's production was a reaction to the M-60.
Obiekt 430 design work continued in parallel and eventually led to the T-64.
Like the M-60, T-62 was an evolutionary, not revolutionary design. It was basically a T-55 with a bigger gun and a longer hull so as to accomodate the longer ammunition of the new gun. Cost of the T-62 was figured as roughly 50% greater than the T-55 which may explain why it was not used by Warsaw Pact armies, even though the Czech's built T-62's locally for export.