"Plymouths and Dodges from the 1960’s. "
While Plymouths and Dodges (and of course, Chryslers) of that era had push-button shifter selectors, those buttons were definitely on the left-hand side of the instrument panel, not on a console.
What Triedag has detailed above is a correct and comprehensive list of vehicles that utilized push-buttons for shift selection.
The OP’s recollection is inaccurate.