How many essentially different strings are there of length $\leq n$ and over an alphabet of size $|\Sigma| = m$? For example, $aaaaaabb \simeq ccccccdd$ essentially, because a smallest grammar algorithm would perform the exact same steps to reduce one as the other.  So how can I phrase this in terms of formal strings, their lengths, etc?
If the length is $0$ there is 1 string.
If the length is $1$ there is $1$ string.
If the length is $2$, then $t = aa$ or $t = ab$ wlog so there are $2$ strings.
$n = 3 \implies t = aaa, aab, abb, aba, abc$ wlog so there are 5 strings!!
How do you expect me to count this without your help :)

The equivalence classes you are trying to count are called "restricted growth strings". The sequence of counts of all RGS of length $n$ are the "Bell numbers", after the mathematician Eric Temple Bell who studied them in the 1930s.
This corresponds to the count of your "essentially different strings" for the case $n=m$. See sequence A000110   in the Online Encyclopedia of Integer Sequences; there is a link to Bell's work in the reference section.
For the more general question, the number of RGS of length $n$ from an alphabet of size $m$ can be computed as the sum of Stirling numbers of the second kind:
 \begin{equation}\sum\limits_{k=1}^m\begin{Bmatrix} n \\ k \end{Bmatrix}\end{equation}
