When the book tells you to upgrade the difficulty of a check does that mean you upgrade one of the difficulty dice to a red challenge die or does it mean you add another purple difficulty die? I am thinking it means the first example here but wanted to make sure.

Transform to red. Upgrade means dice switchery. Increase means more dice.

It can mean both. If there are no purple dice to upgrade, presumably because the pool is full of challenge dice, you add another purple.

As above, the word "increase" always signifies more purple dice.

Page 22 of the core book.

As others have said, upgrading means replacing a / with / . If you don't have any / left, you add a / .