In Rudin's Functional Analysis, in definition 6.3, he defines a family $\beta$ subsets of $\mathscr D(\Omega)$ as follows.
$\beta$ is the collection of all convex balanced sets $W \subset \mathscr D(\Omega)$ such that $\mathscr D_K \cap W \in \tau_K$ for every compact $K \subset \Omega$.
I would understand this definition better if I could think of a counterexample, but I can't. What balanced convex subset of $\mathscr D(\Omega)$ is not open in $\mathscr D_K$, after taking the intersection with $\mathscr D_K$? Is there a simple counterexample for $\Omega = \mathbb R$?
The simplest counterexample $\{0\}$ was given in comments.
Since you asked meanwhile for "a convex balanced set $W⊂\mathcal D$ such that $\mathcal D_K∩W$ is open for some $K,$ but not all $K$", here is one:
$$W=\{f\in\mathcal D\mid f(2)=0\},\quad K_1=[0,1],K_2=[0,3]\subset\Omega=\Bbb R.$$