Suppose $G$ is a finitely generated group which is generated by a set $S=\{a,b,\ldots\}$.
- I don't know terminology here but is there some kind of "orthogonality" or independence that could exist such that if the elements of $S$ are all independent, then $|S|=\operatorname{rank}(G)$?
- If such a notion of independence exists, how is it defined (and how is it called)?
- I believe it would essentially need to be a requirement that none of the elements can be generated by a combination of the others, but is that restriction enough to ensure $|S|=\operatorname{rank}(G)$ or could it still be larger?
- Is there a better definition or even a formula for such concept?
I don't believe any kind of pairwise independence could be used, since if you consider $S_3$, $\{(1,2),(2,3)\}$, $\{(1,2),(1,2,3)\}$, and $\{(2,3),(1,2,3)\}$ would all be S's such that $|S|=\operatorname{rank}(G)$, thus $(1,2),(2,3)$, and $(1,2,3)$ would all be pairwise independent, but then this would be another $S$ only with $|S|>\operatorname{rank}(G)$.
In your case these are different concepts.
Generators are introduced when we analyze the group of rotations, e.g. of a regular n-gon on a plane (dihedral group). So generators are used to show or identify the internal structure of a group, its subgroups for example.
The degree of a linear (=vector space), or rank of a module is the number that gives a numeric value for the structure that we already know everything about. We know that it is a set, which has the scalar multiplication and addition etc.
That is, if you think about the degree of a vector space, a priori you pull a lot of machinery, field or ring axioms, independence of vectors, basis etc. Very specific. You do not need all these for an abstract group.