Well, you can establish the ownership by using the domain tools link suggested in the other thread. That will show you the registration details. Depending on the domain type, it will probably also show you the associated e-mail address.
If you register and pay for a domain, it should be registered to you/your client.
If you take advantage of a “free domain” offer, it appears that sitebuilder.com will register that in the customer’s name, but it would be worth confirming with them first. Some companies register “free” domains to themselves.
Free Domain Entitlement. Free domains are registered on annual intervals on the customer behalf, and will be renewed on annual basis for the lifetime of an account. If a Free domain has been redeemed as part of a website Upgrade and an early cancellation request has been received, Site Builder will deduct the cost of the domain $19.95 from any amount that is eligible for refund. If the refund amount is less than the cost of the domain, Site Builder will not issue any refund to the customer.