This is a fairly straightforward question regarding how we interpret extension of extensions. I just have not come across anyone who has specified this explicitly.
Suppose we have some field extension $K$ of a field $L$ such that $K=L(\alpha)$. Then what how would you regard $K(\beta)$? Would it be $L(\alpha\beta)$ or $L(\alpha, \beta)$?
The correct notation is $L(\alpha,\beta)$, or $L(\alpha)(\beta)$. The notation $L(\alpha\beta)$ is incorrect because it refers to the smallest field which contains $L$ and the product $\alpha\cdot\beta$ (which is sometimes but not always the same as your $K(\beta)$).