@user251222

Hello! Sorry to bother you yet again, but some formal facts are confusing me yet again. It seems to me that the pushout of the diagram $A\leftarrow k[x]\rightarrow k[x,x^{-1}]$ in the category of $k$-algebras is given by $A\otimes _{k[x]} k[x,x^{-1}]$. It's only the coproduct of $k$-algebras that is given by $\otimes _k$.

I think I was originally right. We are taking the coproduct of the diagram in the category of $k$-algebras. However, if we view this as a diagram in the category of $k[x]$-algebras, $k[x]$ is the initial object and we're merely computing the coproduct.

The different is between computing a pushout in $k$-algebras, or a coproduct in $k[x]$-algebras, but these are the same things.