Interpretation: relative clause after a prepositional phrase

Question

Let's consider the following sentence:

He loves these books on a shelf that his dad gave him.

I' m a bit confused about the meaning the sentence conveys. To my knowledge, I think this sentence can be interpreted as follows.

Interpretation 1.

He loves these books, and they are placed on a shelf. These books were given to him by his dad.

Interpretation 2.

He loves these book, and they are placed on a shelf. The shelf was given to him by his dad.

Question.

Which interpretation do you think makes more sense? Is there any general rule as to how we should read relative clauses that appear after a prepositional phrase, or is it just an ambiguous expression that needs more information to narrow down its meaning?

Any comment is absolutely appreciated!

Update.

Actually, I had found some sentences that, I think, give credits to both explanations before I posted the question, but I did not post them at the beginning because I thought it would be so technical that people might be distracted from my question. I decide to put them in the updated post because I am so confused by the structure of these kind of sentences, and they appear so often in math that I need to clarify them in order not to misunderstand.

These sentences come from John Lee's book Introduction to Smooth Manifolds, 2nd edition. Please focus on the sentences with red marks.

(1) (pp. 158) This sentence justifies the interpretation 1.

Torus is not an image of a curve, so the relative clause is not used to modify 'the torus'. In fact, it is used to modify 'the sub manifold H'.

(2) (pp. 206) This sentence justifies the interpretation 2.

'Sense' cannot be continuous in math, so the relative clause must be used to modify 'the curve'.

Answer 1

It's a fairly common example of ambiguity in English. Here's an even more terse example:

Example: There is a bird in a cage that can talk.

Problem with the determiner: The placement of the word ‘that’ confuses the reader as we do not know if the bird or the cage can talk.

Corrected: In the cage is a bird that can talk.

When facing such a sentence, you'd generally use context to infer the intended meaning - so it might be the shelf that was given, but would it make sense to mention that when the speaker only loves the books on that shelf and not the shelf itself?