I am not sure how to answer the question. Most of the answerers have given well reasoned and well supported accounts. It is possible that we have to accept there are different, equally valid, ways of slicing up the language, as long as they are internally consistent.
However, the problem of supposed adverbs is not confined to the copula (the verb to be)
Most, if not all, intransitive verbs can be modified by ‘adverbs’.
“Where are you going?” “Over there.” This might be cheating. My exact meaning depends on my pointing into the distance or at a map.
“How do you feel?” “I feel well.”
“How are you?”. “Well.”
« Comment allez vous? » « Bien »
“Come sta?”. “Bene”.
Even «Τι κάνετε;» «Καλά»
French and Italian have different intransitive verbs for how are you: “How are you going”; “How are you standing”. Greek preserves what in ancient Greek gets called the adverbial accusative of an adjective. Literally, “*What are you doing?” “Good things”! We have a similar adverbial usage of the words good and great. This probably originates from American English.
Here and there are certainly deictic words. This just means they are words of showing (from the Greek deiknumi). The word is equivalent to the Latin-derived demonstrative, familiar to British of my generation.
The sentence “I am here” conveys no information other than that someone is claiming to be somewhere. To have its full meaning, we need to have other information, such as prior knowledge (of the identity of the speaker and, for example, of the fact that s/he has been traveling to a destination already known me). Or perhaps I am a small child and my father is pointing at a spot on a plan he has made of the room in which they both are.
None of this changes what I think is the simple fact that at least for everyday users of the language, ‘here’ is a kind of adverb (a demonstrative, or deictic, if you prefer, adverb). It modifies the verb am, just as well can.