A planetary mass object (also callled a planemo) is an astronomical object large enough to be pulled into a roughly spherical shape by its gravity compressing its matter. A planetary mass object must also have less than about 13 times the mass of Jupiter or about 4,131.4 times the mass of Earth.
If a planetary mass object orbits around the Sun in our solar system it is called a planet (Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, & Neptune) or a dwarf planet (Ceres, Pluto, Eris, Hamaea, and Makemake, plus of number of candidate objects).
If a planetary mass object orbits around a planet in our solar system it is considered to be a natural satellite or a moon. Smaller objects which orbit around planets are also considered to be moons.
Any object smaller than a planetary mass object that orbits the Sun in our solar system is a small solar system body. They include all comets, asteroids, etc. that orbit the Sun ddirectly instead of orbiting one of the planets, moons, asteroids etc. that orbit the sun.
Any astronomical body with a mass greater than about 75 times the mass of Jupiter, or about 23,835 times the mass of the Earth, is a star are the stellar remnant of a star which has completed its "life cycle".
Any planetary mass object which directly orbits a star which is not the Sun, in another star system, is usually considered to be planet. So far there has been no effort to classify exoplanets (planets orbiting other stars) in other star systems as planets or dwarf planets. If they are large enough to be detected they are considered to be explanets. That might possibly change sometime in the future.
Objects which are smaller than planetary mass objects and which orbit around other stars are classified as small bodies in those systems, I guess. Or would be if they were detected. Vast discs containing gazillons of dust particles have been detected in some star systems, but as far as I know no individual objects of less than planetary mass have ever been detected in other star systems.
Any object of planetary mass or less than planetary mass which orits around an exoplanet in another star system would be considered a moon or exomoon. As far as I know there have been only a very, very few alleged detections of exomoons and none of them are confirmed yet. The first exomoons to be discovered and confirmed will probably be very large ones.
Planetary mass objects deep in interstellar space orbiting around the center of the galaxy instead of around any star are called rogue planets. Some have already been detected.
And of course, since small objects are much more common than large objects, rogue planets should be outnumbered many times by rogue asteroids and rogue comets. In recent years a rogue comet and a rogue asteroid were detected passing thorugh our solar system.
So any planetary mass object detected orbiting around any star willbe called a planet or exoplanet. It doesn't matter whether the star orbits around another star or not. A star is very different from a planet, and so an object orbiting around a star is a different orbital class of object than an orbject orbiting a planet.
You might have noticed that the most massive planets would be only about about 13 times the mass of Jupiter or about 4,131.4 times the mass of Earth,and the least massive stars would be at least about 75 times the mass of Jupiter, or about 23,835 times the mass of the Earth.
Isn't it lucky that there are no objects in between that would make it hard to tell the difference between planets and stars.
Actually there are in between mass objects. They are called brown dwarfs. Brown dwarfs have been discovered in interstellar space, and other brown dwarfs have been discovered in star systems with stars.
A number of objects that are discovered have very uncertain estimates of their mass. So in many cases it is uncertain whether they are planets or brown dwarfs, and in other cases it is uncertain whether they are brown dwarfs or planets. But I don't think it is possible to confuse a star with a planet, considering how different their mass ranges are.
And what shoud astronomical objects orbiting around a brown dwarf be called, planets or moons?
According to this planetary mass objects around brown dwarfs are considered planets and not moons.
If planetary mass objects orbiting directly around brown dwarfs are classified as planets, then obviously planetary mass objects orbiting directly around stars must be considered planets with no exceptions.