The classification of verbs into action/dynamic v stative v link/linking is simplistic and can lead to confusion, though the concepts involved are helpful. And note that the action-stative differentiation is semantically decided, whereas the linking-other differentiation is essentially syntactically based, which makes three disjoint classes almost inconceivable.
Nordquist, at ThoughtCo, addresses the stative - dynamic classification cogently [reproduced here with major reformatting and other, minor, adjustments]:
Exceptions: Verbs that 'are both stative and dynamic'
English also has plenty of gray areas, where a word isn't always only
in one or the other category — sometimes words are stative and
sometimes those same words are active. As with so many things in
English, it depends on context.
Sylvia Chalker and Tom McArthur explain: "It is generally more useful
to talk of stative and dynamic meaning and usage [rather than
classes alone].... Some verbs belong to both categories but with
distinct meanings, as with have in
- She has red hair [stative] and
- She is having dinner [active]"
[The Oxford Companion to the English Language Oxford University
Another example could be with the word feel. Someone can
- feel sad (a state of being), and a person can also physically
- feel a texture (an action). They can also tell others to check it out as well:
- Feel how soft!
Or even think can be in both categories, though thinking doesn't seem
like a very dynamic process. Compare the usage in
- I think that's really lousy
with the famous scene in "Back to the Future" when Biff comes up to
George in the cafe and commands him,
- "Think, McFly! Think," while knocking on his head.
When it comes to thinking about link verbs in this context, note that there can be a continuing state involved:
- She was happy with her uneventful lifestyle.
- They remained childless throughout their lives.
or a change
- He became king / angrier and angrier.
- The sky turned grey.
Note also there are verbs fulfilling a linking role but also having real semantic content
- He lay silent on the bed.
- She sat quite still.
- The flags hung limp.
- Resistance proved futile.
- The Roman army emerged victorious.
- The Tallis Fantasia sounded incredible.
- The apple pie tasted delicious.
- George died insane.
- He fell dead at their feet.
- The poor dog was born blind.
- The poker glowed red.
- The soldier jumped clear.
- The idea fell flat.
Again, some of the above examples show stative usages, some show dynamic (and in some cases, it's arguable: is 'glowed' stative or dynamic here?)