Are linking verbs a species of intransitive verbs? Also, is there any difference between linking verbs, stative verbs, and copular verbs?
I would suggest throwing away the terms "linking verb" and "copular verb", unless someone is willing to present you a list of all "linking verbs" or "copular verbs" in English, and a corresponding list of all their grammatical constructions. Those are schoolroom terms, not technical grammar.
That leaves only transitive, intransitive, and stative. Plus be. On those rare occasions when be can be said to have any meaning, its meaning is stative; it is certainly intransitive, and always an auxiliary verb, among its many other peculiarities. I will have nothing more to say about be here.
But most other verbs can appear to be used either transitively or intransively, in one construction or another. As @Billj points out, many stative verbs can be used transitively. Transitivity is a syntactic property, not of a verb, but of the clause the verb is used in. It's hard to say whether walk or eat is categorically transitive or intransitive, for instance, given the evidence below.
- He walked the dog.
- He walked ten miles/a lot.
- He walked too much.
- He walked today.
- Did he walk?
- He ate the steak.
- He ate ten ounces/a lot.
- He ate too much.
- He ate today.
- Did he eat?
Unlike transitivity, which is syntactic and mindless, stativity is a semantic property, and has to do with meaning. However, meaning isn't as simple as one might think. Predicates like verbs can refer to actions (in which case they're not stative) of various kinds; or they can refer to states. However, it's not always obvious which is which, and verbs differ.
For instance, though renting and owning are very similar activities, both part of the Commercial Transaction Frame, rent is an active verb, while own is a stative verb:
- What she said to do was rent that house. ~ He is renting that house.
- *What she said to do was own that house ~ *He is owning that house
Furthermore, there is a tendency (but nothing more) for stative predicates to be adjectives instead of verbs (i.e, verbs tend to be active, adjectives tend to be stative, other things being equal). Likewise, there is a tendency for adjectives to be intransitive; very few predicate adjectives have objects like worth does:
- That picture is worth a large wheelbarrow full of money.
(Although worth has plenty of other peculiarities as well.)
So, to sum up: there are auxiliary verbs in English, quite a few of them; be is one.
Non-auxiliary verbs (the ones with meanings) can refer to actions or to states
(I'm being vague here because both of these terms are complex);
if they refer to actions, they're semantically active, if they refer to states, they're semantically stative.
Grammar terms are confusing. Note that semantically active is not the same as the active in "Active/Passive", which is syntax and has to do with transformations, not meaning.