Those two designs you posted are identical as far as the placement of propellers goes, so yes, they would both work, and with exactly the same flight electronics.
The second ring design though, uses approximately twice as much material for the frame hence it will be heavier, while in practice you'd probably want to minimize the weight of the airframe. There doesn't seem to be a convenient place to mount a central flight computer, but I'm sure a little creativity could lead to a ring design that is practical.
Here's a slightly different configuration for a quadcopter (called V-tail) that seems to be gaining popularity. It would need slightly different logic.