Your router does not have moderate or open nat options.
Options that it does have are Public Lan - nat type and Private Lan -nat type.
Your router also has a DMZ feature and port forwarding options.
User manual for your router is available at http://www22.verizon.com/ResidentialHelp/HighSpeed/Networking/User+Guides/User+Guides.htm
To get your xbox live working properly, you can try using the public nat type, if that does not solve the problem, turn on DMZ for the address of the Xbox360. Now if that still does not solve the problem, then you would need to set up port forwarding for all the ports that xbox live uses.
I would personally recommend that you enable port forwarding for the required ports.
Ports that need to be open are
Together with the user manual and xbox360 portforwarding guides that are available here
You will be able to get it working.
But since that is a lot of information for anyone that has not used port forwarding before I'll write down step by step how to get started.
First get on your computer, You need to connect to the routers own configuration page. By default this is http://192.168.1.1
So, open your internet browser and write HTTP://192.168.1.1 and press enter.
Now you get a a user and password prompt. the username is admin and the password is password. all lower case.
Now on the top menu, click on Firewall settings.
Now select Port forwarding. more information and pictures on the user manual page 98 and forward. You need to create a new rule or rules that forward the above ports - allow them to be open to the internet for the xbox 360.
Looking at page 112. You would create a new port forwarding service, you need to add all the ports one by one. And the host device would be the IP address of the xbox.
Now the DMZ option will be simpler, that is on page 118. You just need to find the xbox ip address from its network options and put it there, and turn on the DMZ. That will allow all traffic for that machine.
Let me know how far you get. I'll help if you get stuck, sorry if this post is a bit unclear, this is a very small box to write in, I can't easily see everything I've already written here.