What if we call it a tiny brown dwarf star?
That would be consistent with the link you posted where they named a Jupiter sized object incapable of fusion orbiting another star a brown dwarf. But I don't like it for two reasons.
Jupiter is in our back yard and has been a planet all this time. I don't really want to be in a two star system, especially if one star is a dud.
Which brings me to the second reason. Stars ought to glow. I think one of the defining characteristics for a star should be that it produces visibile light. Or at least produced light at one time and burned out.