Non-living
Non-living (or sometimes unliving) is a term that refers to all creatures that have never been alive. The term non-living was officially introduced in Shadow of Death manual , where in the description of Elixir of Life (a combination artifact) stated that the artifact "does not work on Undead or non-living creatures". Additionally, the non-undead creatures not affected by the health bonus of Elixir of Life (golems, elementals and gargoyles) have some more immunities in common, which allows to consider them one group of 'non-living' creatures.
Stone and Obsidian Gargoyles are also considered non-living, but they are still affected by morale and mind spells.
Though war machines as units on the battlefield can be considered non-living in some sense, they have their own system of immunities.
Crystal Dragon, despite having a strong reference to the golem-like origin in the description ("made entirely from red crystal and brought to life through magical means..."), doesn't have any features that would distinguish it among all other living creatures.
Non-living creatures
Golems
The only common trait of golems among all other non-living creatures is a spell damage resistance.
Elementals
Air and Storm Elementals
Water and Ice Elementals
Fire and Energy Elementals
Earth and Magma Elementals
Psychic and Magic Elementals
Gargoyles
Immunities
All non-living creatures are unaffected by 25% health bonus of Elixir of Life combination artifact, by morale (except for gargoyles), and the following spells and abilities:
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Archangel's Resurrection ability
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Mind spells (except for gargoyles)
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Unicorns' Blinding ability
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Fangarm's Hypnotize ability
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Creature abilities
Azure Dragon: Fear
Ghost Dragon: Aging
Lich and Power Lich: Death Cloud
Mighty Gorgon: Death Stare
Pit Lord: Summon Demons
Vampire Lord: Life Drain
Wyvern Monarch: Poison
Sea Serpents and Haspids' Poison
Zombie: Disease