Generally speaking, Fidyah is expiation or compensation for infractions committed during the state of ihram: Allah says:

“And perform the pilgrimage and the pious visit [to Mecca ] in honor of Allah; and if you are held back, give instead whatever offering you can easily afford. And do not shave your heads until the offering has been sacrificed;  but he from among you who is ill or suffers from an ailment of the head shall redeem himself by fasting, or alms, or [any other] act of worship.” (Al-Baqarah 2:196)

As Imam Malik says, Fidyah (sacrifice) can be done in Makkah or anywhere else including his home town. As for Hady, it is intended only as a sacrifice to be offered in the Haram to feed the poor. The Hady applies to the following conditions or circumstances:

If someone is prevented from continuing with the pilgrimage after having assumed the ihram as mentioned in the verse cited above.

If someone is performing hajj tamattu` and therefore has released himself or herself from ihram after the `Umrah  before Hajj,

Hady in the case of hunting games while in a state of ihram.

Having said this, however, let me also state that the term Hady can be used as more general and Fidyah to be rather more particular.

Almighty Allah knows best.