"Claim" and "reclaim" have different meanings, don't they?
Yes, but not in this context. It's just a cultural difference.
We also take a lift down to baggage reclaim in the UK, but an elevator down to baggage claim in the US*. There are enough examples of differences between American and British English to fill a book. In fact, this subject has already filled several books.
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*Unless we use stairs or an escalator, of course.
... which one is more correct than the other when it comes to getting our belongings back.
This thread (retrieve vs reclaim) discusses the use of re- as a prefix. Some interpret it as "again" and think reclaim is inappropriate, while re- is also used to mean "back", not "again" - i.e. re-claim is not the same word as reclaim. So both claim and reclaim can mean the same for your bolded quote.