It has been noted by Nancy Levin in her 1980 paper ‘Main Verb Ellipsis in Spoken English’ that copula be cannot be elided under pseudogapping ((2); an example of pseudogapping is given in (1)).
(1) She will enjoy your book more than she will
(2) *She had been happy more often than she had
We can check now whether be of the progressive behaves differently than what is called “copula be” in terms of pseudogapping, i.e. whether the first can be elided under pseudogapping.
(3) She has been enjoying your book more than she has
been enjoying my book
It seems to me that (3) is grammatical, but not being a native speaker of English I can’t be sure. If (3) is indeed grammatical, then the contrast with (2) might show that we should distinguish between a copula be and a progressive be based on the syntactic behavior of the two verbs.
It might be argued that the difference between (2) and (3) is the result not of two different types of be but of the different elements left behind: an adjective in (2) (miserable) and a noun phrase in (3) (my book). But Ed Zoerner and Brian Agbayani argue in their 2002 paper ‘A Pseudogapping Asymmetry’ (http://www.ledonline.it/snippets/allegati/snippets5007.pdf) that adjectives can be the remnant of pseudogapping. Also, it seems to me that pseudogapping of copula be is impossible even when the remnant is a noun phrase (but NP-remnants are possible when progressive be is pseudogapped).
(4) a. *It has been John’s house for longer than it has
been my house
b. They have been living in my house for longer than they have
been living in your house.
In addition, adjectives can be remnants when they are not complements of copula be:
(5) She is wearing blue and I am
Finally, apart from this possible syntactic argument, it has been assumed in the literature that the meaning of the be in (6a) is different than the meaning of the one in (6b).
(6) a. She is a surgeon
b. She is drinking tea
It has been assumed in the literature (e.g. Barbara Partee’s 1986 paper ‘Ambiguous pseudoclefts with unambiguous be’) that the be in (6a) is a function taking as arguments a predicate-type argument and an individual-type argument. This has never been suggested for the be in (6b). If there are indeed two different meanings, then there should be two different lexical items. (See also P Elliott’s answer here: Interchangeable arguments with English copula)
When developing our syntactic parser for English, we at Contextors chose to distinguish between several kinds of be: copular be, progressive be, passive be, among others. Our motivation was mainly practical: we found that making these distinctions helped us manage our rule system more efficiently. In addition, it helped in developing further applications based on our parser, for example a tense recognizer.