Every site is assumed to be under the control of a single author. This makes domain names equivalent to user ids.
You can register a domain name for your public site. If you already have a domain name, you can probably make a subdomain for your wiki site. It need not be on the same computer. That's what's cool about DNS.
I'm registering domain names with name.com. There are many other domain name vendors and most offer many up-sells that make it hard to find what you want. Be patient. Expect to pay 10 or 15 dollars a year.
I feed the DNS system my registered names from a CentOS box running linux. For every domain I've registered, and every subdomain of those, I provide DNS a public IP address. I get this from my connectivity provider.
Wiki can be run in a virtual hosting mode we call a wiki farm. You can direct any number of domains to a farm and they will all have separate sites. I use a DNS Wildcard which gives me unlimited subdomains. For my server in Singapore I added these A records.
$ cd /var/named/chroot/var/named/external/ $ grep asia wiki.org asia A 128.199.166.158 *.asia A 128.199.166.158
See Host Name Resolution for other host name schemes.