A recursive presentation of a group is a one in which there is a finite number of generators and the set of relations is recursively enumerable. I found the following quote in Lyndon-Schupp, chapter II.1:

"This usage may seem a bit strange, but we shall see that if G has a presentation with the set of relations recursively enumerable, then it has another presentation with the set of relations recursive."

It is not clear to me however how one proves it. Does it go through Higman theorem? I.e. one first proves that group with a recursive presentation embeds in a finitely presented group and then one proves that every subgroup of a finitely presented group has a presentation with a recursive set of relations?

And in any case, can one see it somehow directly that having a presentation with a recursively enumerable set of relations (i.e. being recursive) implies having a presentation with a recursive set of relations?