The logical argument for adding a comma before or in the sentence
Yoga is a series of postures, or asanas.
is that a reader who doesn't know what an asana is might otherwise suppose that the term refers to some sort of alternative to a posture (rather than being another way of saying posture). In other words, the reader might read "Yoga is a series of postures or asanas" as including an implied "either" that introduces two incompatible alternatives, much as the sentence "The most feared wild animals in the Sierra Nevada were grizzly bears or cougars" does. The comma before or, assuming that readers understand it as a sign of equivalence rather than as a marker for incompatible alternative options, clarifies that asanas is equivalent to postures, not an alternative to it.
In the example
Joshua was born weighing just 18 ounces – half a kilo or just over a pound.
the writer presents readers with three equivalent measures: "just 18 ounces," "half a kilo," and "just over a pound." The dash following "ounces" stands for something like "that is" and introduces the two equivalents that follow. Although the logic of the "postures, or asanas" example seems to endorse adding a comma after kilo here, the author evidently expects readers to be sufficiently familiar with the word kilogram not to require that the word be spelled out in full—and in the absence of a possible misinterpretation of equivalent terms ("X kilograms" and "Y pounds") as mutually exclusive alternatives, the usefulness of employing the comma to signal their equivalence vanishes.
In the example
By early Tuesday he was dead – a victim of the most deadly of the world’s culinary delicacies, the blowfish or fugu.
as Barmar notes in a comment above, adding a comma before or makes a new misreading possible: that three possible culprits ("the most deadly of the world's culinary delicacies," "blowfish," and "fugu") are under suspicion for the person's death. Of course, that reading is "new" only if the author systematically uses serial commas in lists of three or more alternatives; in AP (or "no-serial-comma") style, the unintended "alternatives" reading is possible when no comma appears before the or.
In my view, the logical argument for setting off equivalent ideas with a comma (as in "postures, or asanas") loses all force in situations involving more than two equivalent terms and no stronger ordering punctuation than commas. But even in instances involving just two equivalent terms, other complications can render strict observance of the "comma before or" rule counterproductive. Consider the third "other example" that Oxford Dictionaries provides:
Spain entered the twentieth century having lost its colonies in the New World and the Pacific in the Spanish-American War or, as it is known in Spain, the War of 1898.
Here, following the "comma before or" rule would produce this punctuation:
Spain entered the twentieth century having lost its colonies in the New World and the Pacific in the Spanish-American War, or, as it is known in Spain, the War of 1898.
But in this case, the intervening phrase "as it is known in Spain" clearly signals that "the War of 1898" is equivalent to "the Spanish-American War"—not the name of a second, separate war. Adding the comma before or doesn't make the sentence easier to interpret; it merely makes a comma-heavy area of the sentence even boggier.
Ultimately, although I recognize the utility of the "comma before or" rule in the "postures, or asanas" example, I think that in many instances the rule isn't useful at all.