I’ve always thought that the “invisibility” aspect of the ring was that it shifted the wearer into the shadow realm. The Nine were invisible without their cloaks, but were visible when the ring was worn. It also made the wearer more visible to Sauron, iirc.

Yup, this is how I think about it: the ring takes your existing point on the scale from the seen/unseen world and inverts it.

So it works out as so:

• Mortal beings without the ring: 100% seen, 0% unseen

• Mortal beings with the ring: 0% seen, 100% unseen

• Immortal beings without the ring: 50% seen, 50% unseen

• Immortal beings with the ring: 50% seen, 50% unseen

For immortals, it doesn't render them invisible because if you "flip" their position, it still basically stays the same.