In Van Morrison's "Into the Mystic" the vocals seem to have an unusual phrasing (for a pop tune). Most other pop tunes I've covered seem to have the melody phrases starting somewhere around the first beat of a measure (give or take a half or whole step). The phrasing in this recording seems to have the melody phrases starting around beat 3 or 2. Is there a name for this type of phrasing? What are some common examples (maybe from the Real Book)? Also, he (the vocals/melody) seems to be hitting some big target notes a beat or even two beats before a chord change. Is this a type of voice leading (i.e. the melody hits a note that foreshadows the chord that we're about to change to)?
I agree with @Tim that the melody starts on the second quaver of the bar. But I don't agree that it is a 3 and a 1/2 beat anacrusis. The reason I say this, is because of the harmony, in other words the chord changes. The vocals aren't an anacrusis to the verse, they start after the beginning of the verse (on the second quaver). The chord changes for the intro and beginning of the verse show this:
First, there is an intro of 6 bars of Eb. Then the main chord sequence for the verse is 8 bars: 4 bars Eb; 2 bars Bb; 2 bars Eb. This is then repeated. It wouldn't make sense to say that the intro is 5 bars of Eb, then the verse is 3 bars of Eb, 2 bars of Bb, 3 bars of Eb. Popular music is usually (not always!) in 2s, 4s, 8s etc.
Although, to be honest, the easiest way to tell that this isn't an anacrusis, is that you can feel the emphasis, of both phrasing and harmony, at the beginning of the bar where the vocals come in, rather than feeling the emphasis on the next bar, which would suggest that the melody starts with an anacrusis.
So, to answer your main question, I think the melody may sound interesting to you because it starts shortly after the beginning of the bar, rather than having a short pick-up (anacrusis). It also feels very slightly "lopsided" because of the 6 bar, rather than 4 or 8 bar, intro.
Anacrusis. Vocals come in on the second quaver of the bar before. So it's a 3 1/2 beat anacrusis. Nowhere near beat 3. Another example is 'Dance On'.