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The rule: "We don’t put adverbs between the verb and the object" (Cambridge Dictionary)

But some sentences confuse me :

1-He drew only a rabbit. (All he drew was a rabbit but he might have done other things while drawing like listening to music)

2-He read only the end of the book. (All he read was the end of the book)

1'-He only drew a rabbit. (All he did was drawing a rabbit)

2'-He only read the end of the book. (All he did was reading the end of the book)

I think (1 and 1') + (2 and 2') are not the same. So are (1 and 2) grammatically correct ? That means rule is not always true ?

To paraphrase "Pirates of the Caribbean", think of this more as a guideline than an actual rule. With creative writing, it is often possible to place the adverb anywhere it sounds good.

Because this is an uncommon placement, when done properly, it can sound dramatic.

They flung wide the doors of the hall, letting sunlight stream into every dark corner.

Done poorly, or in an odd context, it just sounds awkward, e.g. "She ate greedily the cake."

In the future, if you see this kind of sentence structure (and you trust the writer is doing it on purpose) take note of the context, and recognize that the sentence might feel different from the usual phrasing.

[I had already written most of this before @Andrew posted his answer. It says pretty much the same thing, but I thought I might as well post it, having written it.]

"Don’t put adverbs between the verb and the object" is more general advice, probably particularly useful for English language learners, rather than a hard and fast rule.

It definitely is possible to put an adverb between the verb and the object, and often it will sound just fine to an English-speaking ear.

I sang loudly to the audience.

She ran quickly to the bus stop.

The two examples you cite sound fine to me, even if placing the adverb before the verb would sound more 'normal'.

However, it depends on the words being used. As @Andrew says in his answer, it can even be a deliberate choice for literary effect.

As to why some phrases sound complete bizarre, and others sound perfectly ok, I am really not sure.

Eg:

He played brilliantly the piano

sounds completely wrong and would never be said by a native speaker.

Somebody else might be able to explain why.

The rule you've quoted is not always followed. See this discussion, in which it is pointed out that the adverb can go between the verb and the object when the object is long or complicated.

Note, however, that there isn't necessarily a difference in meaning between 1 and 1' and between 2 and 2'. Drawing a rabbit is a little unusual, so let's take the example of eating an apple. "I only ate an apple" means "I ate an apple and nothing else." In contrast, "I ate only an apple" is correct but slightly awkward. In spoken English, you could put an emphasis on the word "ate" to indicate that "only" is modifying "apple" and not "ate", as in:

I only ate an apple.

but in written English (and in spoken English when the word "ate" is not emphasized),

I only ate an apple.

means you ate an apple and nothing else.

If you want to say that you did nothing more than eat an apple, "I merely ate an apple", or "I did nothing more than eat an apple," or, as you suggested, "All I did was eat an apple," would be better ways to indicate that. (The last way is the most natural.)

[1a] He drew [only a rabbit].

[2a] He read [only the end of the book].

With a few minor exceptions the rule you cite holds true.

But the rule is irrelevant in this pair since "only" is not being placed 'between' the verb and its object; rather, it's actually part of the object itself, where it modifies the noun phrases "a rabbit" and "the end of the book", called the 'focus'.

[1b] He [only drew a rabbit].

[2b] He [only read the end of the book].

"Only" usually immediately precedes its focus, as seen in [1a] and [2a]. But it can also be non-adjacent as in [1b] and [2b], where the focus is contained within the VP, meaning that "only" functions syntactically as modifier to the whole bracketed VP.

There is no difference in meaning between [1a] and [1b], nor between [2a] and [2b].