Gaining vigilance any time after the moment you choose to attack with Inquisitor's Ox won't cause it to become untapped.

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.

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.

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.

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.