When you upload a phantom mob skin, there's a button to download an automatically generated texture pack of it. This works for most mobs, but not for phantoms. Phantom eyes have a separate texture at /assets/minecraft/textures/entity/phantom_eyes.png.
The eyes texture is separate from the rest of the mob for an emissive effect, but is not automatically generated by the website. That means when you use the texture pack, it has the vanilla eye textures.