"am/remain/stay" as linking verbs or not

Linking verbs connect the subject to additional info describing the subject (state of being, quality, etc.). As a quick pointer, all forms of be are always linking verbs. (source)

1. I am here. (linking—am is a form of be)
2. I remain here. (linking—remain describes the temporal aspect of where you are)
3. I stay here. (linking—stay also describes the temporal aspect of where you are)

Referencing the source again, if you can replace the verb with am, are, or is and the sentence still makes sense, it is most likely a linking verb.

In your last question, become and seem are, interestingly enough, linking verbs as well. Although here doesn't fit in these cases, the verbs are still linking.

I think the problem is that "Linking Verb" is not defined precisely enough.

All the discussion so far has simply presupposed that there is such a category, and presumed that there was a good definition of it. Somewhere. But nobody's got one that works well enough, so far. Perhaps there isn't one.

"Linking verb" is a sort of translation of the Latin copula 'coupling', which referred to the Latin verb sum, esse, fūī, futūrus 'be'. Everyone agrees that be is an English linking verb. But are there others?

Well, a replacement test won't work. Be is far too idiomatic for substituting it to prove anything.

The suggestion has been made that a linking verb connects the subject to some constituent after the verb, but exactly what kind of constituent, and what kind of connection is made, are not specified.

There are a lot of different kinds of phenomena (constructions and rules, for instance) that connect one part of a sentence with another. Verbs connect noun phrases (like the subject), and other kinds of phrases, depending on the meaning of the verb.

Verbs that can often cause confusion are Locatives, like stand, lie, remain, and fall, as in

• At the top of the stairs stands a grandfather clock.
• The professor's grave lies next to his mother's.
• A lonely mesa rises east of Albuquerque.
• He remained at the wheel for three hours.
• A boundary falls on his property.

Other verbs that can cause confusion are the Aspectual verbs, like start, finish, stop, continue, and remain. Note that remain belongs to both classes; they are not disjoint. Language mostly doesn't have neat boundaries.

• He started/stopped being depressed around November.
• He continued being depressed all year.
• He remained depressed all year.

They take various complement types -- being is required with a predicate adjective for complements of continue, but it's forbidden for remain -- but they're complements, not adjuncts. Aspectual verbs have very little independent meaning; they're close to, but not quite, real auxiliary verbs like be. That's probly why they cause the trouble.

The definition of a grammar school term like "linking verb" (which is taught to children instead of "auxiliary verb" because auxiliary is too big a word) is vague on purpose. It's for children, so it doesn't need to be any more precise than the definition of "bogeyman". Your teachers very likely didn't understand it, anyway, because they'd been taught by teachers who didn't, either, and they, in turn, were also.