The construction you describe seems more like the the category of reductions generated by the abstract rewrite system given by an algebraic theory.
I suggest you take a look to section 8.2("Rewrite systems revisited") of Term Rewriting System where these concepts are defined.
Here a short summary of basic the idea: you can consider a rewrite system as a graph whose vertex are terms of the signature and arrows are step of reduction. From this data enclosing by some operations you get what in the reference is called an abstract rewrite system with compositions, whose objects are terms for the signature and whose morphisms are proof of reductions. Quotienting this structure for axioms of category you get the category of reductions.
Hope this helps.