Given some string u = aardvark over the alphabet {a-z} I want to list the factors. Based on my understanding of the definition of a factor these would be (exhaustively) a, k, aa, rk, aar, ark, aard, vark, aardv, dvark, aardva,rdvark,aardvar, ardvark, and aardvark.
It would not, however, include such strings as d, r, or aaa which are substrings but not factors.
Do the statements above reflect a correct understanding of the definition of factor and substring?
Let $u= a_1a_2\,\dotsm\, a_n$ be a word of $A^*$. A word $x \in A^*$ is a factor of $u$ if $x = a_ra_{r+1} \,\dotsm\, a_s$ for some $r$ and $s$ such that $1 \leqslant r \leqslant s \leqslant n$. This amounts to saying that there exist two words $v, w \in A^*$ (possibly empty) such that $u = vxw$.
Coming back to your example, you gave the list of prefixes and suffixes of the word. The list of factors include other factors, like $rdva$, $d$ or $r$, but not $aaa$.
Reference: see Chapter 1 of [1], the bible on combinatorics on words.
[1] Lothaire, M., Combinatorics on words, Encyclopedia of Mathematics and its Applications 17, (1983), Addison-Wesley Publishing Co., Reading, Mass. New edition (1987), Cambridge University Press, ISBN 978-0-521-59924-5.