Hi, and welcome to the sometimes significantly messier world of small-brand Android devices.
In my experience, LEDison on the Cosmo is good for exactly nothing. I seem to recall that the CoDi was supposed to emulate the five colored notification LEDs of the Gemini. I have never seen that happen though, despite many attempts on various firmware versions. You can use its GUI to create and edit patterns, and you can preview them on the main screen, but that's all, it seems. When I get mail in Airmail, I briefly see the LEDison icon on the OS status bar, but no actual LEDs, neither real nor emulated.
Some individual apps, can flash the single power/notification LED in a single color (on my device, the stock SMS/MMS app flashes in yellow, Skype in green) but that seems to have no relation to LEDison, which seems to just be plainly and completely broken on the Cosmo. At this point, I doubt it will ever get fixed.
The four colored LEDs on the Cosmo seems to have nothing to do with the five (colored) notification LEDs on a Gemini. Two of the four (power and wireless) are present on either end of both the Gemini and the Cosmo, and works much the same. The one on the left side of the device (closest to Q on the keyboard) shows the battery state while charging, though with slightly different meaning to the colors. When a Cosmo is charging, the power LED is red below 80%, then turns yellow and goes green at around 95% charge. As mentioned before, at least on the Cosmo, it can also flash for certain notifications. I don't remember if it did that on the Gemini too.
The wireless indicator mostly stays off on my Cosmo. If I close the lid in flight mode, it turns on in red, when the WiFi hotspot is enabled if flashes slowly in blue. I have seen it come on in magenta under some conditions, perhaps when there's only 2G coverage, but I'd have to look that up to be able to give a more certain answer. One of them (I can't clearly recall which one, though I'd guess it's the power LED), flashes slowly in red when the battery is low.
Edit: typo.