I assume all be verbs are auxiliary verbs; is this correct?
Not necessarily. An auxiliary verb is one which is a verb functioning to give further semantic or syntactic information about the main or full verb following it.
"Be" could be an auxiliary verb i.e. "It has been done"
However, it can also be the main verb itself:
Note 'be' is main verb, and has auxiliary verb "will".
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One of the most common usages of auxiliary verbs is in creating past tenses: I have seen vs. I am seeing vs. I saw
You could not similarly use "run" in the place of "have" or "be". Auxiliary verbs are often called helping verbs--they "help" the reader understand a more specific meaning (that's just to help you remember what it means, not an actual etymology).
Wikipedia gives a list of auxiliary verbs, and it includes fewer than thirty entries including multiple tenses and modalities of the same verb.
There's actually a fairly mechanical way to distinguish auxiliary verbs which does not rely at all on their meaning. Auxiliary verbs always move to the front of the sentence when the sentence is a question. Main verbs never do. If a main verb occurs with an auxiliary, then the auxiliary will move to the front in a question. If the main verb occurs without an auxiliary, then a dummy do will appear where an auxiliary would normally appear.
The examples below show be and have as auxiliaries:
In contrast, the examples below show the same verb as main verbs (and auxiliaries as well):
The reliability of this way of distinguishing auxiliaries from main verbs (in English) is well-supported by quite a large body of linguistics literature. There are further tests which distinguish, e.g., modals from others, etc. You might wish to take a look at the page Modals and auxiliaries to learn more (it's slightly technical, but overall fairly accessible).