IIRC, the deal there is that you can access and use the domain name as if it was your own, but the actual owner remains NameZero until/unless you pay the registration fee directly to Network Solutions (or Register.com, or whoever they are using for the domain name registration process).
What that means is that if you want to move it to another hosting/DNS entity, or if your traffic/bandwidth becomes more than they can handle, they can pull the plug at any point - and you would end up with nothing.
Check the registration for the domain name in about a month or so - Whois.net lists register info for most of the major players. Chances are you won't find your name there (except as the Admin Contact, maybe), but you will find NameZero or their register company as the owner of record.
You should read carefully any provided information about moving the name to another service or supplier. Not saying it's a bad idea, I just wouldn't put a business website under their auspices/control.