(Taking advantage of this week's theme)
Also including the Vermilion Bird in this, which (as I understand it) originated as a separate creature but is often lumped with the fenghuang in particular.
I've recently been fascinated by the phoenix myth, and have been very curious about similar myths (birds associated with fire/luck/immortality/rebirth etc.) from outside Western-influenced culture. Was there any significant cross-cultural contact involving these myths before "modern times" (admittedly fluid...generally meaning pre-significant Western contact, but let's say 1750 for a cut-off date)? Was there any particular reason(s) "phoenix" became the conventional translation for 鳳凰? How did this conflation influence modern interpretations of the creatures?
I know this seems like rambling now, but oh well.
I don't have the expertise to answer the question fully, but I can shed light on certain aspects of the issue. While I do not know when or why fenghuang was first translated as phoenix in English, the first well-known and influential translation of the western phoenix as fenghuang in Chinese occurred in relative recent times. Guo MoRuo, one of the leading literati of early 20th century China, composed a poem in 1920 in praise of the Western phoenix and its rebirth from fire, an attribute never before associated with the Chinese fenghuang. He titled the poem as "凤凰涅磐”，or the "Rebirth of the Fenghuang from Fire". From that point on, phoneix and fenghuang became somewhat conflated in the popular perception in China, even though the educated class still knew the western phoneix by its proper chimese name of "不死鸟” or literally " immortal bird ".
By the way, fenghuang actually refers to a pair of the divine Chinese birds, not one. Feng is the male bird, Huang is the female bird. 凤求凰 (translated as feng courting huang) is a classic music piece of romantic love. Somewhere down the history, feng and huang became conflated with each other to become fenghuang, the female or yin counterpart to the male or yang Chinese dragon.