A polycrystalline diamond compact comprising a diamond table is formed in a high-pressure, high-temperature process using a catalyst, the catalyst being substantially removed from the entirety of the diamond table, and the diamond table attached to a supporting substrate in a subsequent high-pressure, high-temperature process using a binder material differing at least in part from a material of the catalyst. The binder material is permitted to penetrate substantially completely throughout the diamond table from an interface with the substrate to and including a cutting surface, and the binder material is selectively removed from a region or regions of the diamond table by a conventional technique (e.g., acid leaching). Cutting elements so formed and drill bits equipped with such cutting elements are also disclosed.