Mindwrack Demon's delirium triggered ability does not include an intervening "if" clause. This ability triggers at the beginning of your upkeep regardless of the number of types in your graveyard, and it checks that number as it resolves to determine whether you lose 4 life or not.
Because you consider only the characteristics of a double-faced card's front face while it's not on the battlefield, the types of its back face won't be counted for delirium.
The card types in Magic are artifact, battle, creature, enchantment, instant, kindred, land, planeswalker, and sorcery. Supertypes (such as legendary and basic) and subtypes (such as Human and Equipment) are not counted.
Magic: The Gathering · ™ & © Wizards of the Coast · Illus. Matt Stewart
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In some rare cases, you can have a token or a copy of a spell in your graveyard at the moment that an object's delirium ability counts the card types among cards in your graveyard, before that token or copy ceases to exist. Because tokens and copies of spells are not cards, even if they are copies of cards, their types will never be counted.
The number of card types matters, not the number of cards. For example, Wicker Witch (an artifact creature) along with Catalog (an instant) and Chaplain's Blessing (a sorcery) will enable delirium.
