With due respect, where exactly do you see that subdivision in the book?
I'm assuming we're talking about The Linux Programming Interface — A Linux and UNIX® System Programming Handbook by Michael Kerrisk, published by the No Starch Press (SF, CA, USA) in 2010 with the ISBN-13: 978-1-59327-220-3?
I'm not sure what edition I've got (I believe it's the first one), but on Part 7, Chapter 56, which is where Kerrisk introduces sockets, I don't see any such subdivision between "Unix Domain", "Berkeley", "TCP", and "Internet". I was quite surprised to read that claim of yours, because, well, Kerrisk is supposed to be one of the authorities in the field, and I rather assumed that he wouldn't get such a simple fact wrong.
So, re-reading the first pages of the above chapter(s), I confirmed that Kerrisk doesn't give those four options at all. Instead — as expected — he starts by explaining that there is something else in a Unix kernel which allows IPC, using a rather different mechanism than queues on pipes (which he had explained on previous chapters) — sockets.
And he starts by explaining that there are two main domains of sockets, namely, the so-called "Unix Domain" (internal to the host) and the "Internet Domain" (using the Internet Protocol to communicate to the external world), of which there are two subtypes, IPv4 and IPv6.
Besides the domains, there are also (at least) two socket types, known as streaming ("a reliable, bidirectional, byte-stream communication channel") also known as connection-oriented, which on the Internet we usually associate with TCP; and datagram ("[...] message boundaries are preserved, but data transmission is not reliable. Messages may arrive out of order, be duplicated, or not arrive at all.") which we consider to be a type of connectionless communication, usually using UDP.
Kerrisk further explains that "Instead of using the terms Internet domain datagram socket and Internet domain stream socket, we’ll often just use the terms UDP socket and TCP socket, respectively."
I can imagine that your "TCP" classification might have come from this paragraph.
Note that Kerrisk very carefully explains how all of the above are just examples, or specific applications, of certain classifications. Just because the Linux kernel has, by default, three domains for sockets (at least that was how it worked in 2010!), it doesn't mean that all Unixes are restricted to "just three". You can use sockets over other communication protocols, and expose a standard socket interface to the library; the end-user (i.e., the actual developer opening the socket) may not even need to worry how exactly the underlying communication protocol works (it may, after all, be IPoAC, defined in RFC1149 and updated by RFC2549 with QoS, and finally RFC6214 for IPv6) — so long as you use the socket API (and assuming it is well implemented), any transport mechanism can (again: in theory) be abstracted using a socket.
Furthermore, all socket domains, in theory, can use any of the two types, depending on the application (indeed, even Unix Domain Sockets can be used in "stream" or "datagram" mode).
That said, the reference to "Berkeley sockets", IMHO, is merely a historical framework, as mentioned by @duck. One of the many ways to classify the complex Unix family tree is to (roughly) split it between the "System V" and "BSD" ("Berkeley Standard Distribution") flavours, which, again, are the result of the "Unix wars" in the past.