I am working with the following definitions:
If a permutation consists of exactly one cycle (that is, if the $m$ for which $\pi^m = id_X$ is equal to the cardinality of $X$, $m = n$) then we say that $\pi$ is a cyclic permutation.
In the definition above, $X = \left\{1,\ldots,n\right\}$.
If a group $G$ has an element $a$ such that $G = \{ a^1, a^2,\ldots \}$ then we say that $G$ is a cyclic group and $a$ is the generator of $G$.
The name in the first definition is obviously very suggestive, but I'm not sure I can convince myself if the suggestion means anything. The existence of a cyclic permutation does not suggest that the symmetric group of order $n$, $S_n$, is cyclic. Thus my question is, what, if any, is the connection between these two notions?
To be clear, I am well aware that every element of a group generates its own cyclic subgroup, but this fact doesn't seem relevant to the naming convention here.
You asked about the connection between a cyclic permutation and a cyclic group. Cayley's Theorem states that every group is isomorphic to a subgroup of the symmetric group using the regular representation. Thus, in the regular representation of a cyclic group the generator maps into a cyclic permutation of the elements of the group. Of course, you can add extra elements to the permutated set which are fixed by the permutation.
Conversely, given a cyclic permutation. say $\, \pi = (a_1 a_2 ... a_n) ,\,$ the permuted set $\, \{a_1, a_2, ..., a_n\}\,$ can be given a cyclic group structure where $a_1$ is the identity element and $a_2$ is a generator. Multiplication of $a_i$ and $a_j$ is equal to $a_k$ where $i+j \equiv k+1\pmod n.\,$ The permutation $\pi$ corresponds to the generator $a_2$.