I do NOT understand the problem. You pointed one website to another, and this is exactly what happens.
The URL that appears in the browser when you arrive is exactly the URL that you pointed to, as you instructed the webmaster to do.
You pointed ... www.betteorr.com ... to your aw site. When someone types in the www.betteorrcom URL, they arrive at the aw site, with the www.betteorr.com address in the browser rectangle. This is what "pointing" means. Pointing does NOT change the name of one site to another. You cannot point your www.beteor.com to your aw and then expect FAA to reference it as you pointed it. There really is no need for this anyway. You have accomplished what you wanted, by making YOUR OWN domain name the path to your FAA website, which is the FAA url by default.
Your own domain name gives you added credibility for people who find your art by means of using it. It is the URL that you would put on a business card, for example, or on any other promotional material that you might use. The people who get to your art this way would never know that FAA has a default URL towards which you are pointing your own domain name. For FAA, the aw default URL is the address that the coders assigned so that they could even create your FAA website - this is how it is programmed for THIS website. How you use what has already been done for you, then, is up to you.
For FAA and the coders who did all the work to get your aw site up, the FAA default URL is the address. For YOU in your professional presentations, your own domain name in the URL is the address.
FAA made it. You decide how to use it. Does this help?