You have the arrangement right, but the person you're replying to had it right too. It was a massive issue after post-WWI inflation. Reading the wording of the contracts I don't really get it. The contracts say the fare will be a nickle until 1966. But there it is:

> Well before [the price increased to a dime in 1948] the issue of whether to increase the fare had challenged many mayors, become the subject of campaign promises and provoked fierce clashes with powerful interest groups.

https://www.nytimes.com/2019/04/12/nyregion/mta-fare-hike.html

> Mayor John Hylan, who took office in 1918, made the nickel fare a linchpin of his administration and a cudgel he used against the IRT and BRT. The city’s insistence on retaining the nickel fare became a political hot potato that affected every mayor from Hylan to William O’Dwyer, who took office in 1946. During O’Dwyer’s first term, the historic nickel barrier was finally breached, but not before years of contentious, vociferous, and often bitter debates about the merits and problems of charging five cents for a ride that could be twenty miles long from Wakefield in the Bronx to East New York in Brooklyn.

https://www.degruyter.com/document/doi/10.1515/9780823261925-009/html