The situation you describe makes me think of gamification.
Gamification is a powerful tool, but when it's used unwisely, it can ruin a business.
In this case, having multiple ways of rating could lead to confusion. If it had been an attempt to gamify your website, the complex rating would have probably been determined by the internal logic of the system, not by the desire to indulge both lazy users and those willing to make an extra effort.
For you and that site owner might make sense. Now that you've explained the idea behind it, it makes sense even to me, but think of a user who sees for the first time your rating system. All sorts of question will pop out in his/her mind, like:
"Which system of ranting between these two is more important?
Should I use both?
Is it possible to like an item while giving it only one star?"
and so on.
So, instead of making it a matter of alternatives, gamify it:
At first, make it all about the like/dislike rating, which should be the minimal feedback quota the system requires.
Afterwards, you can go in depth by asking the user: How much do you actually dislike this item? And then you can come up with more complex options, while offering the user a badge or some extra points on the website for his active participation in scrupulously evaluating things.
If you are looking for some alternatives, here's a collection of nice inputs:
Is there a better alternative to the 5-star rating system?