Does the Prophet's saying, "It is not righteousness to fast while traveling," mean that it is not permissible for a traveler to fast?

Fasting during travel has three scenarios: The first is if fasting does not cause hardship, then it is better. The second is if fasting causes hardship, then breaking the fast is better. The third is if one is harmed by fasting, then fasting is forbidden and breaking the fast is obligatory. The hadith: "It is not righteousness to fast during travel" applies to the third scenario, where the Prophet, peace and blessings be upon him, saw a man who was exhausted by fasting to the point of being shaded. This means the hadith is specific to those who are harmed by fasting. The hadith cannot be taken in its general sense, because the Prophet, peace and blessings be upon him, used to fast during travel.