You would call something "kaputt" if it can no longer serve its main purpose because of wear or external damage, and if it is more economical to replace that item (or have it repaired by someone) instead of repairing it by yourself.
- a glass jar broken in half
Definitely kaputt, it no longer serves any purpose and must be replaced/disposed.
- a glass jar with a crack that leaks water but not flour
Most probably kaputt, because if the crack is already so severe that it leaks water, it could very easily break altogether in the near future.
- a glass jar with a scratch
"Damaged" or simply "scratched" would be more appropriate, unless the scratch makes the glass unusable, such as with decorative items, or if it poses a health risk (scratch on the rim or handle of a cup).
- a glass jar with the lid missing
A missing lid can of course be reason for replacement, but neither the glass nor the lid is kaputt.
- a book, partly burnt but readable
- a book with some pages ripped-out
- a book with a folded page
Those are various states of being damaged. "Kaputt" would mean that the book is completely unusable, such as when its back is broken with all the pages loose, or complete destruction by fire or water.
- a computer that doesn't respond to anything
If it can be "repaired" by a reboot, definitely not kaputt. If the damage is permanent, such as in a hardware failure, then it's kaputt. Of course, that also depends on how tech-savvy the user is. A complete novice can perceive a computer to be kaputt, whereas a more experienced user with the right tools might not.
- a computer that without an OS
Not kaputt, because all it needs is a new OS.
- a keyboard with a stuck key
Only if it's a somewhat important key, or if it can't be easily repaired. If it has to be replaced, it's kaputt.