I just tried both of the apps you mentioned and you’re right, their media widget stays even after being interrupted. By far, I think Pocket Casts is doing the best job of maintaining the widget and closing it only after an episode is done.
I looked into the issue and a couple of users are describing the problem I’m facing. I also found this forum post with the same problem Android 13 update causing issues with lock-screen and menu "miniplayer"