Church-Turing Thesis - the mechanical model - the turing machine- its limits and its equivalence with a modern digital computer
Modern computers are not more powerful than a TM.
The meaning/definition of &quot;mechanical means&quot; is not &quot;a TM&quot;; &quot;mechanical means&quot; refers to any step-by-step process that can be carried out in the real, physical world.  For instance, it would include mechnical devices, electronic devices, a human following a procedure &quot;mechnically&quot;, and so on.
Wikipedia does not define &quot;can be calculated by an effective method&quot; as &quot;can be calculated by a TM&quot;.  It doesn't define &quot;mechnical&quot; as can be calculated by a TM.  Rather, it says that the Church-Turing hypothesis asserts that any computation that can be carried out mechnically, can be carried out by a TM.  That's not a definition of what it means to carry out a computation mechnically -- that's a consequence of the Church-Turing hypothesis.
I think you are getting confused about the difference between a definition of computation, vs what the Church-Turing thesis claims/asserts/conjectures to be true about computation. The Church-Turing thesis is not intended as a definition of computation; it's intended as a statement/claim/assertion about computation.  The Church-Turing hypothesis doesn't provide a formal definition of &quot;effective computation&quot; or &quot;mechnical means&quot;; it leaves that up to the intuition.