If a Brown Dwarf has or had fusion, is it a star or a dead star?

Is a Brown Dwarf a star or a planet? Is a dead star a planet? If it was formed like a star and had some fusion, isn't that different than the formation of a planet? If it doesn't have fusion, it's not a star, but what if it once had it? They say that at 13 to 65 times the mass of Jupiter, brown dwarfs can fuse deuterium and lithium. Does that make it a star?

They say that the smallest star is the red dwarf OGLE-TR-122b. Our Sun is 1,048 times the mass of Jupiter. I'm just a noob who is not especially good at maths, but this is what I looked up:

1.988920E+30 kg, The Sun
0.189860E+30 kg - 0.198892E+30 kg, OGLE-TR-122B (100 times Jupiter to .1 solar mass)
0.129280E+30 kg, Brown Dwarf (lithium fusion)
0.123409E+30 kg, Brown Dwarf (deuterium fusion)
0.001898E+30 kg, Jupiter

The smaller star, OGLE-TR-122B, is estimated to have a radius around 0.12 solar radii, or around 20% larger than Jupiter's, and a mass of around 0.1 solar masses, or approximately 100 times Jupiter's.

What is the smallest star?

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