These days, every shared hosting company, including GreenGeeks, offers a "free domain name" or "free domain name for life" promotion on sign up. They both are referring to the same thing. As long as you are paying for their web hosting services, they will waive the annual registration fee for a domain name through their registrar. So theoretically, if you do pay for the web hosting services for life, you indeed can get a "free domain name for life."

When signing up for this free domain though, I would not recommend registering your main domain this way. It is not safe to assume the web host will be around forever, or that you will want to stick with the same web host for a long period of time. Transferring your domain might turn out to be a major headache. Web hosts have been known to impose a domain name transfer fee, should you request to take your domain name with you to your new hosting provider. Normally, this fee is reasonable (in the $20 dollar range), but there could be other fees on top of that.

If you really value your domain name, the smart thing to do would be to register with a respectable 3rd party registrar. It is not expensive (GoDaddy does this for $10/year per domain). Use the free domain offer to register a domain you will not be devastated to lose.