For Tri Delta, at least in my chapter, the president's pin is owned by the chapter and is lent for the current president to wear during her term. Our chapter's president's pin was a gift from executive office (or maybe the foundation?) when we were chartered so I'd assume each chapter's president's badge is a bit different as the years have gone by.
It is a bit larger, flatter and with blue stones compared to the standard badge.
Standard badge:
Couldn't find a picture of my chapter's president's badge, but it is similar to the Field Consultant badge: