If you really do not want your personal information to be visible with 100% guarantee, you need to ask someone else do the registration for you (on their name, but then you will have to make pretty sure that the link between you will stay strong as if they are problems on the domain name, the other party is its owner, not you; for this, companies sometimes use their attorney when they want to do some "secret" registrations, for example before opening new services).
Because otherwise you have only two options:
- if the TLD in which you do the registration offer "private registration" (which is often the case for European ccTLDs when individuals register domain names), then opt for it, but the registry will have your personal data… just that it will not be displayed through whois
- or, use your registrar (or a third party) proxy/privacy service where, basically they put their names as owner instead of yours
(Have a look at https://en.wikipedia.org/wiki/Domain_privacy and one of the latest effort to enable this privacy mode by some known people at https://njal.la/ )
In all cases you should double verify at least:
- in case of problems of your domain names (such as third party requests for take downs with DMCA, or ICANN procedures such as UDRP or URS), what happens? Is the "cloaking" immediately reverted? How are you contacted?
(have a look at https://www.cnet.com/news/private-domains-not-so-private/ for an example of such stories)
- even when everything goes right, are you still reachable (email, phone, fax, postal address) through the intermediary visible in whois?
- what kind of contract do you have with the party using its name instead of yours? Because in the second case, the registry will know only the third party, not you, and if there is a complaint on the domain, this third party will have all rights on the domain name, you will have none; so you should really check the contract between them and you, to make sure the domain name could not be stolen from you.
When you use the third party proxy/privacy service, you should of course double check how they operate. It is kind of strange if the domain name is first registered with your info and then later on cloaked. But it may be so, have you asked your registrar/provider about its service on this topic? If you have seen your personal information on line for the specific domain, it may be too late indeed. Depending on why you wanted this feature, you may now go shop for another domain name (and as hinted above, the rules depend on the TLD, ccTLD especially European ones will have stronger protection on personal information, so shop carefully).
You should also always use the registry authoritative whois server, preferably on the command line, not any kind of web whois.