Has this generalization of Turing machines studied before?
A general notion of machines based on graphs was given by Wilfried Sieg and John Byrnes in K-Graph Machines: Generalizing Turing's Machines and Arguments. It may or may not be related to what you are doing.

Also note that it is quite easy to find a complete formalization of Turing machines. Why are you saying that "the tape of a Turing machine ... is usually left out when introducing Turing machines". That's an odd thing to say. Let's perform an experiment. For instance, what is unclear about the definition given on Wikipedia (following a standard textbook)? All you need is a formalization of finite sequences, and I'd be flabbergasted if Mizar hasn't got those.