These are two different representations that depend on your assumption. Some linguists believe that noun and determiner make a Noun Phrase, while others believe that noun and det constitute a Determiner Phrase. The second group assumes that in such languages as English determines are overtly presented while in such languages as Russian they are covert or null. It is a rough explanation. Let's be a little bit more specific.
According to the X-bar theory every head must project into a phrase. A determiner does not seem to be a phrase, thus the representation in (1) causes a trouble. That's why many linguists reject the representation in (1), where the det is a specifier of an NP. On these grounds, the representation in (2) is theoretically more acceptable. It was firstly proposed by Abney (1987).
Now, let's look into some empirical evidence toward 2. Here are two genitive phrases:
- Helen's book
- The girl with a telescope's book
In the first example, it looks like the genitive suffix attaches to the noun head, however the second example clearly shows that it is not the case. That genitive suffix is actually attached to something bigger. It attaches to the whole phrase and it is actually a determiner:
[DP Helen [D' 's [NP book]]]
It is also impossible to say Helen's the book.
In introduction courses many instructors prefer to use the first representation in order to keep things simple.