WebID is the correct form according to the w3c spec.
Public ID and Public Name are both used to refer to the same thing, which is in part the SAFE version of a clearnet ‘domain name’, but also used as a personal identifier on the SAFE Network.
At this time only the Public Name/Public ID is part of the official SAFE API. Support for a SAFE WebID is experimental but I think likely to become part of the official API at some point and in some form.
What are they?!
Here’s my understanding, which may need improving 
Clearnet
The function of a domain on the clearnet (as an address for Internet services such as a website) is different to a WebID (which is an identifier for individuals or groups, such as an organisation), and can be used by those entities to verify themselves, establishing the right to access data or services. So a clearnet WebID is a kind of universal login, with some extra capabilities built in, such as additional information about the owner entitiy(s) (eg like a public profile).
SAFE
On SAFE, a Public Name (aka Public ID) handles both functions: it is an address for services such as a website, and an identifier which can be used to establish rights of access to data and services (ie by being logged into the account which owns that Public Name). So it covers much of what both a WebID and domain do on the clearnet, but without the extras such as a public profile. Those extras could have been added in a SAFE only way, but by adopting the form of a WebID we get the extras, plus quite a bag of bonuses.
A SAFE WebID establishes compatibility with a powerful feature that has the support of Tim Berners-Lee and others, and brings with it an ecosystem of applications, tools, and data (ie Solid, the Semantic Web/Linked Data, SPARQL queries etc) that can change the way everyone relates to information, applications and services, in ways that will add enormous value to the SAFE Network.
For anyone who wants to learn more about that value, may I recommend my presentation to the SAFE DevCon in April 2018: