The Magic Initiate feat allows you to pick a 1st level spell, which you can cast once per long rest. From the PHB (p. 168):
Choose a class: bard, cleric, sorcerer, warlock, or wizard. You learn two cantrips of your choice from that class’s list.
In addition, choose one 1st-level spell from that same list. You learn that spell and can cast it at its lowest level. Once you cast it, you must finish a long rest before you can cast it again using this feat.
Your spellcasting ability for these spells depends on the class you chose: Charisma for bard, sorcerer, or warlock: Wisdom for cleric or druid; or Intelligence for wizard.
The spell is not prepared.
The Sage Advice Compendium addresses this when discussing choosing a spell with the ritual tag on page 3:
If a druid takes the Magic Initiate feat and chooses detect magic as their one spell, can the druid cast that spell as a ritual?
A druid’s Ritual Casting requires a ritual to be prepared. The spell from Magic Initiate is known but not prepared.
The character just gets one casting per long rest. The other Sage Advice passages quoted in the question address casting the spell using spell slots which would require the caster to prepare the spell.