I think a few issues need to be cleared up first.
Firstly, the first source you mentioned is an ESL source rather than a linguistics source. If you'd like to learn more about linguistics, I'd suggest consulting more linguistically-oriented sources like Payne's article, since ESL sources tend to be simplified - they are, after all, aimed at language learners, not linguistics students.
Secondly, another issue to be cleared up is whether we're talking about descriptive categories, which are specific to languages or comparative concepts, which are used for cross-linguistic comparison. It seems here that we're talking about descriptive categories, but you did not specify this in your question, so I think it is helpful to make this explicit. This is because the discussion of what constitutes a lexical verb and a copular verb in English has nothing to do with, say, French être or Mojave iðuːm, or even Middle English be.
Finally, Payne never defines the lexical verb as a copula verb. He is, in fact, arguing against the treatment of the copular verb be as an example of a lexical verb. He regards it as an auxiliary, and uses several kinds of evidence to show that the copula be and the auxiliary be behave the same way morphosyntactically except when the difference in behaviour is purely semantic/pragmatic.
More generally, lexical words are generally those that form an open class in the language, i.e. a class of words to which words can be added freely. For example, in English, you can easily form new words, e.g. you can form new lexical verbs by X-ise meaning 'to make something like X'. It's much harder to form new auxiliaries like can or may. This is the 'general idea' behind the distinction.
However, these criteria seem not to be really helpful for determining whether be is lexical or auxiliary in its predicative use, since we don't know whether be belongs to the first group or second group just by looking at it. This is where morphosyntactic criteria come in: We can define lexical verbs and auxiliary verbs by means of their morphosyntactic behaviour. This is the whole idea behind language-specific descriptive categories.
Descriptive categories are invented on a language-specific basis to describe the particularities of the grammar of a particular language, in this case English. Payne cited language-internal evidence for treating the copula and auxiliary be both as auxiliary, e.g. be can take the negator not and be fronted in questions, whether as a copula or as an auxiliary for tenses:
(1) a. Was he a good boy?
b. Was he crying?
(2) a. He is not a good boy.
b. He is not crying.
He then suggests that the traditional opinion that the copula be is a lexical verb is based on the erroneous assumption that every sentence must have a lexical verb.
So Payne a) suggested a language-specific definition of lexical verbs , and b) defended the unified treatment of copular and auxiliary be as both auxiliary.
