I would like to know why we define a space $X$ to be semi-locally simply connected if
$\forall p\in X \exists U\ni p: i(\pi_1(U))=0\subset \pi_1(X)$ (SLSC),
where $i$ is induced by $U\hookrightarrow X$. My question is: why isn't the definition
$\forall p\in X \exists U\ni p: \pi_1(U)=0$ (SLSC')
sufficient? Of course I can imagine neighbourhoods of a point for which this distinction makes a difference. However, I cannot imagine any space where there is a point for which no environment has vanishing $\pi_1$, but there is an environment with vanishing image of $\pi_1$ in the fundamental group of $X$.
The Wikipedia page on semi-locally simply connected has a good example of a space that is semi-locally simply connected but not locally simply connected: the cone over the Hawaiian earring.
The thrust of this example is that small open neighborhoods will generally contain non-contractible circles—but these circles can be contracted in $X$ by collapsing them to the cone point.
This space still satisfies your second definition, which is a bit weaker than locally simply connected, but this can be fixed, for example by gluing another Hawaiian earring to the cone point.