This semester I was introduced to finite automata (FSM), then pushdown automata (PDA), and now the Turing machine (TM). Granted that there're many possible implementations of these abstractions (electrical, mechanical, whatever...), I find the terms used in describing these machines rather extravagant. It's as if the terminology is meant to be agnostic of any specific implementation, but ends up just reflecting the time and the area of application of its provenance. Hence references to tape, read/write head, and the like.
I'm wondering if I could better understand the essence of and the differences among the FSM, PDA and TM if these were cast in terms of the basic computer architecture of today. It would seem that an FSM would require, at a minimum, a read-only memory (ROM) to implement branching (jumps), acceptance or rejection, as well as register(s) to store the input and compare it to the finite states loaded in ROM. By contrast, a PDA adds a rather peculiar kind of writeable stack-based memory allocation to store the input for later processing. Finally, the TM relaxes the stack-based restrictions on the implementation of RAM.
Is this an accurate description of the three machine types? Does such a description lose in explanatory power to the traditional presentation in terms of states, tape etc.? Is such a description of necessity more complicated and adds extraneous detail compared to the bare minimum necessary to understand these concepts in the traditional presentation -- beyond the fact that these terms, too, will be rendered obsolete and are application specific?