Can this card be used to claim any card placed into the discard pile (whether played for its effect or discarded) or is it just for those discarded for an extra die or when drawing 2, keeping 1?
All cards placed in the discard pile are discards, regardless of how they got there. The rules say to shuffle the discarded cards - it doesn't make any distinction about how they got there. So you can take any card regardless of why it is being discarded.
