The meanings of a verb when used as an auxiliary are almost always different from those when not used as an auxiliary. Instead of trying to consider them all the same word, I find it easier to treat the different meanings of a word as different words, as if they're homophones. This is useful for words whose different meanings have different grammar, and "be", "have", and "do" are excellent examples of that.
Wikipedia has a nice table of auxiliary verbs in English. Here's what is has for "be":
be1 | copula (= linking verb) | She is the boss.
be2 | progressive aspect | He is sleeping.
be3 | passive voice | They were seen.
These three uses mean different things: the copula (#1) links a subject to its identity or characteristics, the progressive aspect (#2) forms a progressive tense of a different main verb, and the passive voice (#3) puts the main verb into the passive voice.
As the Wikipedia article notes, some people consider the copula to be an auxiliary, in which case it's both an auxiliary verb and the main verb, and others consider auxiliary verbs to only be verbs which are not the main verb. With this in mind, you can check to see which definition is being used. The example you had found clearly treats the copula form of "be" as not an auxiliary, so it considers main verbs to never be auxiliary verbs.
I find "have" to be an even clearer example of different meanings, because "be" and "do" are more complicated. There's an auxiliary version of "have" that's used to put verbs into perfect tenses. This is pretty different from the other verb "have" which means to possess something. You can see this in sentences like "I have(AUX) had(MAIN) many enemies."
"Do" also has different meanings when it's an auxiliary or not. When it's an auxiliary, it either has essentially no meaning, and is just required for grammar reasons, or it adds emphasis to the statement. When it's the main verb, it has a long list of different meanings and uses. Think of "I did(AUX) do(MAIN) the homework", or "Did(AUX) you do(MAIN) something about it?"