The sentence states that all the children got presents and the fact that they were the luky ones only shows an <em>additional information about them</em>, that's why it is placed <em>between commas</em>; this is called appositive. The sentence is also nonrestrictive because <em>it doesn't say</em> that only the children who were lucky got presents, <em>but all of them</em>.
The correct options are "The sentence contains a restrictive clause", and "Only the lucky children got presents".
A restrictive clause is <u>the use of an adjective to modify a certain word inside a sentence</u>. This modification has to be <u>crucial in determining the meaning of the entire phrase</u>.
In this case, the word "lucky" acts as the modifier for "the children". It's a restrictive clause because it tells us that only the lucky children got presents. The word "lucky" is restricting the intended meaning of the phraseby excluding unlucky children.