Short answers, pulled from the story.
A fusor is a proposed astronomical term for any object capable of nuclear fusion in its core during its lifetime. The term was proposed by Gibor Basri and is more inclusive than "star" because it covers both active stars and many brown dwarfs.
Gibor Basri proposed the term to the International Astronomical Union. His definition covers any object that achieves core fusion at any point during its existence.
The minimum mass is roughly 13 Jupiter masses, the point at which deuterium fusion becomes possible. This is significantly lower than the roughly 75-80 Jupiter masses needed for sustained protium, or ordinary hydrogen, fusion.
A fusor is any object capable of core fusion, while a planemo is a round object that never achieves core fusion. A planet is then defined specifically as a planemo whose primary orbit is around a fusor.
Many brown dwarfs qualify as fusors under Basri's definition because they achieve deuterium fusion during their lifetimes. The definition only requires that fusion occur at some point, not that it be sustained.
Hydrostatic equilibrium is the stable state in which the heat generated by core fusion halts an object's gravitational contraction. Objects reach this state and become main-sequence stars at around 75-80 Jupiter masses, when inward gravity and outward thermal pressure balance.