http://creativecommons.org/licenses/by-sa/3.0/
Kinsella suggests using the share-alike license. I posted about that here as well.
That is for books though, I’m not sure why it might differ for websites.
As of the moment I simply have a non-copyright statement, saying isn’t copyright. But I fear this is NOT the solution. Would this allow someone else to simply come in and claim the rights to my website?
No, assuming you pay for your own hosting and own your domain. You need to be careful to renew your domain registration, but really only if someone is being predatory to “steal” it. More likely, you have a small number of visitors with no interest in messing with you.
See the article, Man who received speeding ticket acquires city police domain name