The 2006 definition of a planet states that a planet is a celestial body that
(a) is in orbit around the Sun
(b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium shape
(c) has cleared the neighbourhood around its orbit.
Point c is why Pluto allegedly is no longer a planet. However some "proponents" of the definition, namely those who use logic, figured out that according to this vague definition, if we take it literal and set a arbitrary border on "clearing the orbit", then either Pluto is still a planet or we have no planets at all. Therefore these guys actually set up an own definition, thereby replacing point c by the requirement that a planet must be the dominant mass in its orbit. This is not what the 2006 definition is telling, so these anti-Plutoers aren't actually proponents of the 2006 definition. They have their own definition according to which really only Pluto is no planet because Neptune dominates its orbit.
However their definition still doesn't exclude Eris from planethood. Its orbit is far from any of the eight recognized planets. From a vertical point of view, Eris' orbit intersects with that of Pluto. However Eris' orbit is highly tilted to the ecliptic and the two bodies don't really come close enough that their gravities would influence each other strongly. Eris is more massive than any Kuiper belt object (including Pluto). Eris itself is not really a KBO because it only intersects the Kuiper belt while most of its year being outside it. So there is no reason to ban Eris from planethood if point c states (according to some anti-Plutoers) that a planet must be the dominant mass throughout its orbit. So why don't these guys consider Eris a ninth planet?