There's a continuum of sorts between gas giant planets and dwarf stars, with a few notable points where you could draw the distinction. They all come from the same general start - a cloud of interstellar gas collapses into a spherical object. Depending on how big it is, you can get different objects.
First you have gas giants, no fusion at all. This would be your Jupiter and Saturn type planets. Jupiter is actually about as big, volume-wise, as a gas giant can get. Add more mass, and it starts getting denser rather than bigger.
At 13 Jupiter masses, you have enough gravitational pressure to fuse deuterium. This is what most astronomers define as a brown dwarf star, but others, and apparently you, consider it to still be a planet. Previous terminology included "substar", which I would not be opposed to. Deuterium isn't particularly common, so these objects glow very dimly, as far as stars go.
At 65 Jupiter masses, you can start fusing lithium as well. This is one way to distinguish brown dwarfs from other stars - red dwarfs and yellow dwarfs, like our sun, consume their starting lithium very quickly, and so the presence of lithium spectra indicates a brown dwarf.
At around 80 Jupiter masses, it starts fusing hydrogen, becoming a red dwarf, like Proxima Centauri. Still very dim, but at this point it's undeniably a star.
At around 750 Jupiter masses, the star develops a more complex internal structure, and becomes a yellow dwarf, such as Sol.
So where do you draw the line? Anywhere you want, but most astronomers settled on the simplest one: if it's undergoing fusion, it's a star, if it isn't, it's a planet.