Does Freedom of Movement prevent damage from Spike Growth?

Question

The D&D 5E spell Spike Growth includes:

The ground in a 20-foot radius centered on a point within range twists and sprouts hard spikes and thorns. The area becomes difficult terrain for the duration. When a creature moves into or within the area, it takes 2d4 piercing damage for every 5 feet it travels.

The spell Freedom of Movement has:

You touch a willing creature. For the duration, the target’s movement is unaffected by difficult terrain, and spells and other magical effects can neither reduce the target’s speed nor cause the target to be paralyzed or restrained.

The question is, with the target of Freedom of Movement being unaffected by difficult terrain, their movement speed is unaffected by the difficult terrain from Spike Growth. Are they also unaffected by Spike Growth damage?

Answer 1

No

As stated in spike growth:

When a creature moves into or within the area, it takes 2d4 piercing damage for every 5 feet it travels.

The trigger for taking the damage is the creature moving into or within the area. This is not contingent on the area being difficult terrain for them.

Answer 2

No

Spike growth creates two effects: difficult terrain, and damage. It's not the difficult terrain that is dealing damage; it is the spike growth that is dealing damage. Freedom of movement does not grant immunity to spike growth.

Freedom of movement does not state that its target would be unaffected by damage, so it is affected normally.