Independent set poly reducible to Strongly independent set?
Assume that the integer $k$ of the (decision version of the) independent set instance $G$ you are reducing from is larger than $1$ (the case $k \le 1$ is trivial).
Let $G'$ be the graph of the corresponding instance of strongly independent set.

As it is written in your solution notes, all the additional vertices are connected to one another with a clique. If you select any of them, say $w_e$, then you cannot select any other vertex $v$ in $G'$ since the distance in $G'$ between $v$ and $w_e$ is at most $2$. This implies that the size of a maximum strongly independent set of $G'$ is $1$. Therefore, if there is a strongly independent set of $G'$ of size $k$, it must not include any vertex $w_e$.

The correctness of the reduction is then easy to see:


If $S$ is an independent set for $G$, then $S$ is also a strongly independent set for $G'$ since all pairs vertices in $S$ are at a distance of at least $2$ in $G$ and hence they are at distance of at least $4$ in $G'$.
If $S$ is a strongly independent set for $G'$ with $|S| \ge k$, then $S$ cannot include any vertex $w_e$, and is therefore a subset of the vertices of $G$. The distance between any two distinct vertices in $S$ is larger than $2$ in $G'$, and hence it is larger than $1$ in $G$.