A little Wikipedia history on Public Ivies: the term apparently started in a 1985 book that listed UVM as one of 10 Public Ivies. (Not necessarily Ivy-equivalent in prestige, but in education quality and feel.)
A 2001 book by other authors suggested 30 Public Ivies and didn’t include UVM as one of them. Today UVM is somewhere in the 30s as far as USNWR ranking of public universities (as is Binghamton).
So people don’t think of UVM as a Public Ivy that much anymore, but they do have that 1985 book to point to, as I’m sure their PR department is aware.