The wedge sum of two pointed types $(A,a)$ and $(B,b)$ can be defined as the higher inductive type with the following constructors:

Points come from the sum type?$in : A + B \to A \vee B$

And their base point is glued $path : inl(a) = inr(b)$ Clearly this is pointed.

The wedge sum of two types $A$ and $B$, can also be defined as the pushout of the span

$A \leftarrow \mathbf{1} \rightarrow B$

where the maps pick the base points of $A$ and $B$. This pushout is denoted $A \vee B$ and has basepoint $\star_{A \vee B} \equiv \mathrm{inl}(\star_A)$