Because depending on the context of the story a character dying and finally being free from some kind of pain can be seen as a happy ending and I have a nagging feeling that that isn't what this group was made for.

2170948 One easy system would be:
If things get worse in the end it's a bad ending.
If things improve in the end it's a good ending.
Though, by this system the story ending you are describing would indeed fit in the good ending category.

Do you have any better idea for a system of determining if a story has good or bad ending?

What about bittersweet endings?

To me, at least, a happy ending is where the situation has improved from the start, and the main problem that's causing the plot is solved, and usually the main character gets in a better position than they were in the beginning.

2193740
It depends how you define 'bittersweet'. If it's a bittersweet ending like, say "the villain is defeated/the problem is solved, (maybe) things will change for the better, but the main character is probably traumatized by what he/she went through and several characters died, like the ending of Harry Potter, it's allowed, because the ending was happy, it's simply that the cost to get there was high.
Really, we need a thread to define 'bittersweet' as well.

2170948
That's allowed. It's a happy ending because they're not in pain anymore. And, as Albus Dumbledore says:
"Death is but the next great adventure." And, unless your story is about the Princesses or Discord or similar, the characters are going to die eventually, even if not on screen.