That’s a good rule of thumb… but it’s probably not enough; no reasonable definition would call Jupiter a star, or even a brown dwarf, or the Solar System a binary system, yet the Sol - Jupiter barycentre is outside the sun… (the whole system’s barycentre is sometimes inside the sun, but that’s due to Saturn’s, Uranus’, and Neptune’s pulls cancelling Jupiter’s).

I’d call the barycentre thing a necessary but not sufficient requirement; a proper definition of double planet should probably also take into account other factors like the relative mass and density of the bodies, and their minimum and maximum distance.