Yeah, your recollection is wrong. Other civs' actions do not affect the faith cost of your Great Prophets (or other Great Persons).
Thats not quite right! The cost to found a pantheon keeps going up, as other pantheons are founded. Likewise, the cost to found a religion (i.e., the cost for your first GP) goes up as other religions are founded. The cost for subsequent GP is unaffected by other civss actions.