Let $G$ be a finite group. Call a presentation of $G$ "normalised" (I do not know whether such presentations by generators and relations have been studied before and I invented the name "normalised" here) if it is of the form
$$\langle x_1,...,x_n \mid x_i^{r_i}=1, \ i=1,...,n ; v_t(x_1,....,x_n)=w_t(x_1,...,x_n) , t=1,...,u\rangle. $$
Here $x_i$ are generators (so that we can not omit any $x_i$) with $r_i \geq 2$ their order and the $v_t$ and $w_t$ are words in the $x_i$ of length at least two (We really want them to be words in the $x_i$ that do not contain inverses of the elements $x_i$).
Examples:
$G=\mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$ has the normalised presentation $\langle x_1 ,x_2 \mid x_1^m=1 , x_2^n=1 , x_1 x_2=x_2 x_1 \rangle$.
The standard presentation for the dicyclic group ($n=2$ gives the quaternion group) $\langle x_1 , x_2 \mid x_1^{2n}=1 , x_2^{2n}=1 , x_1^n= x_2^2 , x_2 x_1 x_2^{-1} = x_1^{-1} \rangle $ is not "normalised" since in $x_2 x_1 x_2^{-1} = x_1^{-1}$ there appear inverses.
Question: Is there a standard procedure to obtain a normalised presentation for a given finite group (it seems they always exist)? Can one add additional hypothesis on the "normalised" condition to make such a presentation unique with some special properties (one nice condition might be that all generators have the "smallest possible" orders)?
It would also be interesting whether it is possible to obtain normalised presentations using GAP. It would also be interesting to know for which finite groups $G$, there exists such a normalised presentation where all generators have order two. Maybe someone knows a reference for that.