There are several versions of the definitions of the FST and the Mealy machine. Some of the definitions are almost same. Some have a little differences. But it seems that they both are a kind of DFA with an output function $G \colon \Sigma \times Q \rightarrow \Gamma \times Q$, where $\Sigma$ is the input alphabet and $\Gamma$ is the output alphabet.
I'm not sure about the differences between them.
Are they the same machines? Why is there two different names?
BTW, I found that there is not the set of final states in the definition of Mealy machines or Moore machines. On the other hand, final states do exist in the definition of FST, although Mealy machine is one type of FST. Does it mean that final state is not important when we are talking about the Mealy machines?