Ann Hogue's First Steps in Academic Writing contrasts linking verbs with action verbs, and the same terms are used in a number of other books, including McGraw-Hill's essential ESL grammar by Mark Lester. This use of the word action could easily lead to a confused contrast with stative. On the other hand, maybe another terminology is partially at fault. ESL Beginner by Johanna Pugni, Linda Robbian, Sherry Boguchwal, Dianne Ramdeholl says linking verbs are sometimes called state-of-being verbs. Or perhaps the confusion is preexisting and the terminology is the result. Whatever the source may be, there's a problem.
Linking verbs are those verbs that allow adjective phrases as one of their complements. These include words like be, look, seem, feel, sound, get, become, stay, etc. Even where these verbs take noun or preposition phrase complements, they are generally still linking verbs.
- He seems nice/a good fellow/up to it.
- It was good/a pen/under the bed.
- She became healthy/president.
In contrast, stative verbs have nothing at all to do with complementation. A simple rule of thumb is: if you can't put it in the progressive, it's stative:
- it seems nice -> *it's seeming nice stative
- I have a car -> *I'm having a car stative
- It gets hot -> It's getting hot dynamic
- It works -> It's working dynamic
It may be coincidentally true that many linking verbs are stative, but the two sets of properties are orthogonal.
When it comes to teaching, the stative/dynamic dichotomy is most relevant when teaching progressive aspect. In contrast, verb complementation is typically best dealt with at the pattern level as a verb is learned. It may be useful to bring it up, however, when addressing passive voice because students could mistake an NP complement for an object and try to front it with ungrammatical results.
- she became president -> *(the) president was become by her