Who are those who enter Paradise through the gate of Ar-Rayyan? Are they those who die while fasting, or those who fast even if they die while not fasting, or those who die during Ramadan?

The meaning of "the fasting ones" in the Hadith (definition) refers to those who frequently fast in this worldly life, not to someone who died while fasting or in Ramadan (definition). The gate was named "Ar-Rayyan" because it is derived from Ar-Rayy (quenching thirst), which is fitting for the condition of those who deprived themselves of thirst in this world. They enter through it to be safe from thirst and are given a pure drink before entering Paradise. This gate is exclusive to them, to the exclusion of others, due to the hardship of thirst they endured in their fasting.