All objects in the following live in an abelian category. Consider the short exact sequence $0\rightarrow B\rightarrow C\rightarrow D\rightarrow 0$. Apply the Hom-functor $\text{Hom}(A,-)$ where $A$ is also an object in that category. This gives a left exact sequence $$0\rightarrow \text{Hom}(A,B)\rightarrow \text{Hom}(A,C)\rightarrow\text{Hom}(A,D).$$
Now I extend the resulting left exact sequence by appending a morphism $\psi:\text{Hom}(A,D)\rightarrow\text{Ext}^{1}(A,B)$, establishing exactness at $\text{Hom}(A,D)$. I know I can further extend this method to extend further by using the functors $\text{Ext}^n(A,-),n\in\mathbb{N}$ appropriately, but this is not the main point of this question.
I do know the interpretation of $\text{Ext}^{1}(A,B)$ as the group of equivalences classes of short exact sequences $0\rightarrow A\rightarrow E\rightarrow B\rightarrow 0$ with an object $E$ from this category with $A$ being a subobject of $E$ and a surjective morphism from $E$ to $B$ respecting the exactness of the short exact sequence, two such short exact sequences being equivalent if there is an isomorphism between the middle terms (indicated by $E$).
How does the interpretation of $\text{Ext}^{1}(A,B)$ as (i) the group of equivalence classes of short exact sequences of the form $0\rightarrow A\rightarrow E\rightarrow B\rightarrow 0$ relate to the interpretation of $\text{Ext}^{1}(A,B)$, (ii) which says that $\text{Ext}^{1}(A,B)$ measures the non-exactness of the Hom-functor at $\text{Hom}(A,D)$.
I am completely stuck. I do not see the connection between the two views. Clearly the definition of $\text{Ext}^{1}(A,B)$ as a group of equivalence classes of short exact sequences, using the Baer sum as a binary operation seems mathematical rigorous enough. The second view is also a bit sketchy through my description, which also needs clarification.
I am thinking that if $\text{Ext}^{1}(A,B)=0$, then, as a first illuminating but incomplete approach to the question, this left exact sequence can be extended to a short exact sequence as well, but even this seems unclear to me; why does the morphism from $\text{Hom}(A,C)\rightarrow\text{Hom}(A,D)$ become surjective in that case?