Editor Alert....those are for titles, not proper names. As the rule quoted above says, "books/films/songs/games". Those are titles of a specific work, which are also always appear in quotes or italics (also as shown in the quotes above).
I would say proper names always have capitals (but see below).
For instance, "The Beatles" always appears as the Beatles in text.
That is not true; writers vary. In particular, when the group was established, and people referred to each member as "a Beatle", then "the Beatles" referred to the group of individuals. But, as I said before, the text on the sleeves of the albums that have text refers to "The Beatles" throughout, as do some (not all) of my books on them.
Also, what you quoted was the Oxford style guide, which is a collection of style rules specific to Oxford University. It's not an overall grammar book.
It's not a matter of grammar, as I pointed out before; every press and university, and anyone else, can choose to define their own style; but there are certain common conventions which are almost entirely identical whichever body's style guide you choose to look them up in. To illustrate the difference, consider the following from the Chicago University Manual of Style, which disagrees with the Oxford one on this particular matter:
Q. I would like additional clarification regarding a recent Q&A. You stated that a word that is part of a proper name should not be capitalized if such word is being used as an article in the sentence. The example given was “We evaluated the University of Texas’s enrollment data.” You stated that this was correct even if the proper name of this school is “The University of Texas.” I thought that the goal of good editing was to produce clear, accurate, and comprehensible text. If “The” as part of a proper name is not capitalized, the reader will be led to believe that it is not part of the name. Why create confusion, to say nothing of the insult that may be given in certain cultures and communities where a name may have great significance?
A. Chicago’s style is to lowercase, probably because it’s not practical for editors to research the capitalization of “the” in every organization name in a manuscript. Even quick Internet searches leave questions, since Web sites often show inconsistencies in spelling and capitalization. That’s the idea behind having a style manual in the first place: to make arbitrary decisions (which we would not call “correct,” since there are many acceptable styles) that an editor can impose efficiently within a document for the sake of consistency. Readers are not likely to be insulted if the name of their organization is treated the same as that of every other organization in a document.
Note: "That’s the idea behind having a style manual in the first place: to make arbitrary decisions (which we would not call “correct,” since there are many acceptable styles)".