This is only really true since we usually define an algorithm to be that which is computable by a turing machine or an equivalent model. It makes sense to talk about algorithms that run on turing machines with oracles attached, but these may not be computable by a turing machine alone.
I'd express this more like:
"A Turing machine is a very simple abstract machine designed to capture our intuitive understanding of computation in the most general sense. Any formal system that can simulate a Turing machine, and thus also perform arbitrary computations, is called Turing complete. (Untyped) Lambda calculus is Turing complete."
Extremely minor quibble re. this footnote:
This is only really true since we usually define an algorithm to be that which is computable by a turing machine or an equivalent model. It makes sense to talk about algorithms that run on turing machines with oracles attached, but these may not be computable by a turing machine alone.
I'd express this more like:
"A Turing machine is a very simple abstract machine designed to capture our intuitive understanding of computation in the most general sense. Any formal system that can simulate a Turing machine, and thus also perform arbitrary computations, is called Turing complete. (Untyped) Lambda calculus is Turing complete."