Confusion about transitive vs. linking verb

I am having trouble understanding transitive, intransitive and linking verbs:

Can a linking verb / "verb of incomplete predication" ever also be considered an intransitive or transitive verb, or will it always be labeled a linking verb? (For an exercise in which students must label the role of each word).
Am I correct to believe that all verbs must be either transitive, intransitive or linking verbs? Or are there more categories?

Relatedly, can a word ever be both the object of a transitive verb AND the complement of a linking verb? I am okay with identifying whether a verb is transitive or intransitive, but I get confused differentiating between
a) a transitive verb with an object
b) a linking verb with a complement
c) a transitive verb with a complement (is this even possible? or would it always be called the object?)

ex. He told a lie.
Would "lie" be the object of a transitive verb
or the complement / predicate noun of a linking verb?