Yes. Plus the turing machine has an infinite memory tape to write and read. Something that is in scope of mathematics, but we don’t have any infinite tapes in reality. That’s why we call it a mathematical model and imaginary… and it’s a useful model. But not a real machine. Whereas an abacus can actually be built. But an Abacus or a real-world “Turing machine” with a finite tape doesn’t teach us a lot about the halting problem and the important theoretical concepts. It wouldn’t be such a useful model without those imaginary definitions.

(And I don’t really see how someone would confuse that. Knowing what models are, what we use them for, and what maths is, is kind of high-school level science education…)