I tried to find an example in which "consecutive" make a difference, but I can't find it. For example here I have an odd number of exchanges regardless of whether I limit myself to consecutive or not:
And here I have an even number of exchanges independently I limit myself to consecutive interchanges or not:
Can you give me a counterexample in which there is a difference between making exchanges in general or making exchanges only between consecutive ones make a difference in odd/even steps we need to reach the final state?
A such counterexample doesn't exists because the parity of a permutation doesn't depends on the decomposition into product of transpositions (adjacent or not). This means that, although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same.
Consequently, we don't need the “consecutive” term in this definition.