For the Conjure Fey spell, who chooses the CR, whether it's a beast or fey, and the actual creature - the player, or the DM?

Question

The description of the conjure fey spell says (emphasis mine):

You summon a fey creature of challenge rating 6 or lower, or a fey spirit that takes the form of a beast of challenge rating 6 or lower.

I would like to know what the player actually gets to choose:

• Does the player choose the CR, whether it is a beast or a fey, and the creature?
• Does the the player choose the CR and whether it is a beast or a fey, while the DM chooses the actual creature that appears?
• Or does the player only choose the CR, while the DM chooses the creature and whether it is a beast or a fey?

Answer 1

The DM chooses everything

The Sage Advice Compendium deals with this in the question "When you cast a spell like conjure woodland beings, does the spellcaster or the DM choose the creatures that are conjured?" (p. 16).

It states that "Some spells of this sort specify that the spellcaster chooses the creature conjured" and gives Find Familiar as an example ("an animal form you choose"). And also "Other spells of this sort let the spellcaster choose from among several broad options" giving Conjure Minor Elementals as an example ("You choose one the following options for what appears").

The spell description is explicit in what elements the caster can choose. If it is silent, the choice falls to the DM.

For Conjure Fey the description gives no choices to the player, therefore everything is up to the DM.