The Chinese Phoenix represents the Empress and is the second-most respected creature, second only to the Dragon. The Phoenix is the leader of all birds. the phoenix is called hō-ō and also goes by the name fushichō ("immortal bird"). In Japan, the phoenix goes by another name, the Suzaku/Red Bird. One of the Shijin, the four guardians of the compass. Each guardian is associated with a direction in the compass. The Suzaku is the guardian of the south.
It is debatable whether the Suzaku/Red Bird is simply the Japanese version of the Phoenix or if both legends are merged together to make one.
Basically... The Suzaku and the Phoenix are essentially just the same thing.
The answer to your thread title question is hō-ō. Just like the Pokemon. Ho-oh.