Connecting The Informal and Formal Definitions of Decidable With Each Other
The author himself wrote informally speaking, the second is not a definition in the same sense of the first.

Intuitively, since a recursive function is defined by either $\mu$-recursion or composition of simpler functions, the existence of a total recursive function provides an effective method to compute the answer.

The informal part of the above reasoning is that we didn't really define what we mean by "effective method". 

The usual formalization involves the definition of a computational model, which provides a rigorous framework for the notion of algorithm. Once you have done that, you can prove the equivalence in a mathematical sense.