Ostensibly different products on Ext-groups

Show that each of those products distributes over the others, and use Hilton-Eckmann (over a field, or for $A$ projective; in general, I don't know...) This ends up being then an exercise in using naturality.

It can be done more concretely, too. For example, to show (1) and (2) are the same, one can show that if $E$ and $E'$ are $n$- and $m$-extensions of $k$ by $k$, then $E\circ E'$, the Yoneda composition, is equivalent as an $(n+m)$-extension, to $E\otimes E'$; this also shows (1) is (3), because why I wrote $E\otimes E'$, an iterated extension of $A$-modules, is really the result of restricting scalars along the coproduct of $A$ from the iterated extension $E\otimes E'$ of $A\otimes A$-modules. On the other hand, to show directly that (2) and (3) are the same, show that computation of both in terms of the bar resolution is actually the same.

Later. Let $E$ and $E'$ be $n$- and $m$-extensions of $A$-modules of $k$ by $k$. For example, $E$ is $$0\to k\to E_{n-1}\to E_{n-2}\to\cdots\to E_0\to k$$ I'm going to grade this complex so that $E_0$ is in degree $0$, and write $\bar E$ for the result of simply taking away the rightmost $k$, so that $\bar E$ is a complex with homology $k$, concentrated in degree $0$. Then $\bar E\otimes\bar E'$ is a complex of $A\otimes A$-modules, which, by the Künneth formula, has homology $k\otimes k$ in degree $0$. Since in degree $n+m$ the complex $\bar E\otimes\bar E'$ has $k\otimes k$, we can add the $k\otimes k$ to it in degree $-1$ and obtain an $(n+m)$-extension of $k\otimes k$ by $k\otimes k$ in the category of $A\otimes A$-modules, which is what I denote above $E\otimes E'$. If we restrict scalars along the diagonal map, we get a complex $\Delta^\*(E\otimes E')$, which is the product of $E$ and $E'$ accoding to your product (3).

The other important observation is the following: what I wrote $\bar E\otimes\bar E'$ is a double complex with the shape of a rectangle. By looking at it mildly hard, you can see that if you go from the module in the largest degree (which is $k\otimes k$) towards the one in degree zero by walking along one side and then along another, you get a simple complex which is more or less trivially isomorphic to what now I would write $\overline{E\circ E'}$, the truncation of the Yoneda composition (if you walk along the other two sides of the rectangle, you get $\pm\overline{E'\circ E}$, and pursuing this you get a very concrete proof of graded-commutativity) Finally, you have to notice that there are maps of extensions $E\circ E'\to E\otimes E$ (and $\pm E'\circ E\to E\otimes E'$, of course)

Sorry if this came out rather messy: it is the kind of things that maximally adapts to an explanation face-to-face in front of a blackboard!

That (1) and (2) coincide is also shown here in a general nonsense manner.