Suppose $G=\langle S\mathrel| R\rangle$ is a presentation of a group (or semigroup) such that:
- $G$ is not generated by any proper subset of $S$.
- For any $r\in R$ we have $\bigl\langle S\bigm| R\setminus\{r\}\bigr\rangle\neq G$.
Does such a presentation of $G$ have a name? (Note that I understand this presentation need not be unique, and maybe even need not exist, but it does exist for my $G$ and I don't want to reinvent the wheel. I also know that "minimal" is used when speaking about finitely generated groups, but mine is not one.)
I think the word you are looking for is aspherical.
There are (classically) 5 different definitions of an "aspherical presentation". They may all be stronger than what you need though; I give this as an answer because your presentation may be aspherical (and certainly being aspherical implies your property). The reference is: Chiswell, Collins and Huebschmann, Aspherical group presentations, Mathematische Zeitschrift 178, 1-36 (1981) MR0627092.
As a warning: this paper clarifies certain issues in one of the standard tests in geometric and combinatorial group theory (Chapter III of Lyndon and Schupp Combinatorial group theory). so do not look up that text!