Just because verbs can have both transitive meanings and intransitive meanings does not mean that transitivity is about meaning not grammar. Because having object complements is a syntactic function of a verb, transitivity is principally a matter of grammar. It’s about how some bits of the sentence relate to other bits through their orderly systematic arrangement, and that is within grammar’s domain.
What arguments you give a verb determine that verb’s valency. Wikipedia’s article on this opens:
In linguistics, verb valency or valence is the number of arguments controlled by a verbal predicate. It is related, though not identical, to verb transitivity, which counts only object arguments of the verbal predicate. Verb valency, on the other hand, includes all arguments, including the subject of the verb.
And under transitivity they continue:
In linguistics, transitivity is a property of verbs that relates to whether a verb can take direct objects and how many such objects a verb can take. It is closely related to valency, which considers other verb arguments in addition to direct objects. The obligatory noun phrases and prepositional phrases determine how many arguments a predicate has. Obligatory elements are considered arguments while optional ones are never counted in the list of arguments.
Traditional grammar makes a binary distinction between intransitive verbs that cannot take a direct object (such as fall or sit in English) and transitive verbs that take one direct object (such as throw, injure, kiss in English). In practice, many languages (including English) interpret the category more flexibly, allowing: ditransitive verbs, verbs that have two objects; or even ambitransitive verbs, verbs that can be used as both a transitive verb and an intransitive verb. Further, some verbs may be idiomatically transitive, while, technically, intransitive. This may be observed in the verb walk in the idiomatic expression To walk the dog.
A transitive verb is one that can accept a direct object argument. It does not require that there be one. As you mention, many verbs that accept object arguments can also be used without those arguments with a nuanced meaning distinct from the version with object arguments. Such verbs are sometimes called ambitransitive.
The thing is, these verbal arguments occupy particular syntactic slots. Consider these three sentences using the ditransitive verb bake respectively with one, two, and three arguments:
- Your sister bakes.
- Your sister bakes a birthday cake.
- Your sister bakes you a birthday cake.
That third sentence has a verb and three noun phrases as that verb’s arguments: subject, indirect object, and direct object.
Your sister[subject] bakes[verb] you[indirect object] a birthday cake[direct object].
The indirect object obligatorily falls immediately after the verb without any other constituent intervening between those two. It is ungrammatical to say:
- Your sister bakes *always you a birthday cake.
The grammar forbids it. This is therefore a restriction not of meaning but of syntax — and thus of grammar proper.
Similarly, the order of the indirect and direct objects must be strictly observed. It is also ungrammatical (in standard English) to place the direct object before the indirect one:
- Your sister bakes a birthday cake *you.
This too is a matter of grammar, not of meaning.
Now consider what happens when we use dative alternation to produce a sentence equivalent in meaning but whose grammar now differs:
- Your sister bakes for you a birthday cake.
Now there is no indirect object at all, only a preposition phrase connecting to the recipient, which was the old indirect object. Moreover, unlike with indirect objects, that prepositional phrase is now movable:
Your sister bakes a birthday cake for you.
For you your sister bakes a birthday cake.
So sentences 6, 7, and 8 all have the same meaning as sentence 3, but a completely different grammar. The rules about what you can and cannot do are different between those two sets, despite them all meaning the same thing.
This shows that a verb’s arguments are about more than meaning alone. It’s about grammar.
Although I have proven this point only for ditransitive-vs-transitive verbs instead of for transitive-vs-intransitive verbs, I hope you see that verbal arguments are about grammar not about meaning.