The motivation for the spelling rule (which the Woodward English site does not specify) is that we don't want adding "-ing" to change the length of the final vowel (the one before the consonant). A single vowel before a single consonant at the end of a word is usually short, but a single vowel before a single consonant before another vowel is usually long. Thus, adding "-ing" to a word like "stop" without doubling the consonant would result in the spelling "stoping" (which is a different, and rare, word), and this would indicate that the vowel was long. Since this would violate the "no vowel length change" rule, we double the consonant ("stopping"), which indicates that the previous vowel should remain short.
Vowels that are already long don't need the spelling change. Even after adding "-ing" ("eating", "hating"), the vowels will remain long. As you may know, the fact that a vowel is long in a word such as "eat" or "hate" is often indicated by a silent vowel, either immediately after the first one (like the "a" in "eat") or separated from it by a single consonant (like the "e" in "hate").
Meanwhile, in words where there is a final "e", but the vowel is short anyway ("live", "have"), the shortness of the vowel is so programmed into an English speaker's head that adding "ing" without doubling the consonant ("living", "having") doesn't cause a net change in the way we pronounce the word. That's why rule (1) on the Woodward page says that if the verb ends with an "e", you drop it and don't double the consonant. The same principle applies to words such as "head", where there is a silent vowel following the spoken one, but the initial vowel is short. Thus, "heading" is also pronounced with a short "e".