I ordered a customized phone via edge.
It was complicated and most folks didn't know how to process it. I guess since it's not an iPhone 6, Verizon did the bare minimum in terms of "training".
When you go into the store (and presumably over the phone), Verizon will have two listings for the 2nd gen Moto X. One for the 16GB ($499) and one for the 32GB ($549). If you're doing Edge, or even Edging Up, your agreement will be for either model. Once your agreement is processed, Verizon sends the order to Moto, and the Moto will send you a pin directly to your email. I got mine within a half hour from leaving the Verizon store.
You'll then go to Moto Maker and select the option of "I have a pin" or whatever it's called, it should be recognizable.
When you enter the pin, the site will know if it's good for the appropriate memory size you selected. I don't know what happens if you select 32 when your pin is for a 16 and vice versa (experiment at your own risk).
I then proceeded to "order" my phone via Moto Maker after I entered the pin, at the end, because I chose a Wood back, Motorola charged me $25+tax for that customization. I think it's not straight forward that you pay Verizon for the memory upgrade but not for the back cover upgrade, but whatever. If you're not changing the back, and you're simply customizing the color scheme, your order should come out to $0. It may still need a credit card, I am not sure.
Once you've submitted your order, Moto will let you know how long before you'll get your phone.
I order a 32GB on Friday and it says expected deliver of Oct 22. I thought it was because of the wood so I decided to change it to regular black plastic, but the delivery date didn't change, so then I figured if I'm going to wait just as long, I might as well get the Ebony. I think the delay had to do with the memory size, and not the materials. Another member of the forum (JustinHEMI) ordered a 16GB and he said his expected deliver was the 8th. My brother also ordered (albeit a different provider) a 16GB and he mentioned his delivery was expected on the 8th as well.
Also, Verizon will be singing me a pre-paid box/bag for me to mail the old Moto X, which I won't do until I have received and activated the new one.
That's the best I've got for you! Good luck!