Apologies if the title is confusing, but I couldn’t think of better phrasing in short text.

Whenever we define / specify a certain automaton (such as a finite state machine or a turing machine) by defining all of its States, transition function, etc., this feels awfully similar to defining an algorithm. For example, I can define a machine that can tell if a number is divisible by 3. It is very similar to writing an algorithm and the steps to solving the problem.

Now I understand that the two aren’t exactly equivalent. But would it be incorrect to say that the specification of a machine is a type of algorithm, since we’re defining the steps it takes to solve a problem (how to respond to a specific state or input to solve a specific problem)?

Yes, that’s essencially the church-turing-thesis

https://en.wikipedia.org/wiki/Church–Turing_thesis

That’s the only church I’ll trust