The usual diagram for the homotopy extension property is:
where $i_t^X:X\rightarrow X\times I,x\mapsto(x,t)$. Isn't this the same as saying the following square is a pushout? $$\require{AMScd} \begin{CD} A @>{i}>> X\\ @V{i_0^A}VV @VV{i_0^X}V\\ A\times I @>>{i\times \mathrm{id}}> X\times I \end{CD}$$ If not, why not, and if so, why isn't this the standard way of stating it?
When talking about homotopy extension, there are more requirements than when talking about a general pushout. It is not true that $X\times I$ is the pushout of $X$ and $A\times I$.
For example, let $A$ be a point, and let $X$ be two points. Then the pushout of $X$ and $A\times I$ is the disjoint union of an interval and a point, whereas $X\times I$ is two intervals.