If you're able to start up the software after having entered your key the first time, then you are using the Full Edition. The Full Edition won't start without a valid registration key. UNLICENSED will appear in the 'About ArtRage' window if a name wasn't entered with the key, but that's just the default text so it's cosmetic, it doesn't change the functionality of the software at all.
A quick way to check if you're using the Full Edition - press 8 on your keyboard to select the tube paint tool and try to place some on the canvas. If you're using the Starter Edition, nothing will happen, that tool only exists in the Full Edition.
You can force ArtRage to ask for your information again in order to re-enter the name and key by holding down the TAB key as it starts up, but if you're already running the Full Edition, the only change that will occur after doing this is that UNLICENSED will change to the name you entered.
If you have any problems, feel free to email me at email@example.com and I'll be able to help.