It can't hurt to try and replace the LEDs as Espeelark suggested (or just remove the ones that didn't have resistors) and see if the lighting then works again. Ideally, as in an ideal circuit the LEDs that had the resistors should have been protected by those resistors and should still work
if the common (+ blue wire) has not been damaged. Without knowing how Ring does protection, if any, on the LM it could be possible that you've killed whatever's supplying the common in that very brief overcurrent moment between when you turned on the LEDs without resistors and when the LEDs went out.
As for how the outputs work, they are switched ground, with the common (+v) always present on whatever you've got hooked up to the outputs. I imagine there is only one source supplying this so if you've killed this source then that's it no more outputs (unless you put in your own external bridge rectifier and hook it up to the blue wire

).
- Tim