I'm a neuroscientist by training, and that doesn't match my understanding (although the definitions might have changed).

Inflammation would be an instance of nociceptive pain, and an NSAID would alleviate it by reducing the inflammation; and most tissue damage will result in inflammation.

Neuropathic pain, on the other hand, would be due to damage of the nerves themselves, and NSAIDs are completely useless here (ask anyone with sciatica or other nerve entrapment)