I prefer the Day/Nite mirror in whatever variety because it is wider and offers more visibility, and the benefit of knocking down the retina-burning glare of that freaking Jeep with projector bulbs behind me.
While not a "65 expert", as Woodchuck noted, unless your car had the Visibility Group, it would not have come from the Factory with a Day/ Night mirror. However, one could have been Dealer Installed.
From what I have seen on 64's and early 65's, what I call the "Rocker Type" (picture attached) is what would have been furnished with the Visibility Group. Possibly this is what you call the "push button" ? I base this use particularly on an unrestored , January 1965 Mustang on another Forum.