The idea of transitiveness is a subset of the technical definition of verb-valency in linguistics.
When we delve on the subject of verb-valency, we should no longer use the term "transitive verb". We should walk on a higher plane, with more grandiose pedestals and say "transitive usage of a subject of a verb". Alternatively, "transitive dependencies of a verb." A similar topic of concern is found in transitive dependencies and valencies of a database/information entity.
What qualifies as subjects worthy of transitive action of a verb might still be in trivial dispute.
He came eating.
Is eating qualified to be the transitive-subject of come?
He came yesterday.
Why would adverbial use of yesterday not qualify as valency of a verb??
She gave me.
And yet me qualifies as valency?
He came groaning.
Why wouldn't groaning increment the valency of came?
She gave a cake.
AFAIAC, all subjects above qualify as transitive-subjects of its verb, not intervened or mediated by any preposition. Some people may disagree with me, limiting themselves to the initial thesis on linguistic valency-of-verbs. Perhaps, I am contorting the intents of that thesis. Perhaps, current people indulgent in that thesis attempt to concoct an algorithm to restrict the term valency, without considering the mathematical implications of such restrictions.
Valency 0:
- Giving is such a blessing.
Valency 1: One action-subject to the verb in a phrase.
- Giving presents is a blessing.
- He bakes.
The perpetrator is the sole subject of the verb.
Valency 2: Two action-subjects per verb. Valencies 1 & 2:
- Giving me(v1) presents(v2) is his delight.
- He(v1) bakes cakes(v2).
- He(v1) baked yesterday(v2).
- He(v1) baked laughing(v2).
Valency 3:
- He(v1) giving me(v3) presents(v3) is his delight.
- He(v1) bakes me(v2) cakes(v3).
- He(v1) baked cakes(v2) groaning(v3).
Valency 4:
- He(v1) giving me(v3) a kiss(v3) yesterday(v4) was amusing.
Valency 5:
- I am amused when he(v1) gave me(v3) a kiss(v3) groaning(v4) yesterday(v5).
Test of valency: A subject increases the valency of a verb, when that subject can be used with the verb independent of any other valency candidates.
- he(v1) gave me(v3) a kiss(v3) groaning(v4) yesterday(v5).
- he gave
- gave me
- Q: When did he give you a kiss?
A: He gave yesterday.
- Q: How did he give you a kiss?
A: He gave groaning.
- He gave a kiss