Is a register machine built out of automata of some sort?
I can only react to this as seen from the perspective of a theoretical computer scientist.
A finite state machine/automaton is, as you know, usually represented by a finite directed graph. The vertices of the graph repesent the states of the automaton, the edges are the instructions: in state $p$, upon reading symbol $a$, move to state $q$.
Next to this graph is the &quot;machine interpretation&quot;. The machine has a control unit and an input tape. That control unit is represented by the finite graph.
From these finite state machines/automata we can build more complex automata by adding external memory. For instance the pushdown automaton. The external memory is a single stack. The edges/instructions of the finite state repesentation are now extended with tests and instructions of the external pushdown: test what is the topmost symbol $A$, pop $A$ and push a new sequence $\alpha$.
The register machine likewise is obtained by choosing another external memory type. That of registers. The machine has a finite, fixed, number of registers each holding a natural number. Each register can be tested for zero, incremented and decremented. The instructions of the register machine are added to the finite state control. So we get edges/instructions of the form: in state $p$ upon reading $a$, when register $i$ is nonempty, increment it by $1$ and move to state $q$. If we allow the register machine to read $\varepsilon$, that is compute without reading input, these machines are Turing complete (even with two registers).