The first strikes me as rather uncommon, and has more of an idea of entertaining, lecturing, pontificating. The person holding court is the centre of attention, but actively.
if you are in the public eye, you probably are, or have become, simply a well-known figure. So, royalty, politicians, actors, footballers, etc are in the public eye.
If you are in/under the spotlight, the attention is more temporary. Price William is always in the public eye, but his coming marriage has put the spotlight on him and Kate Middleton. Being in the limelight is similar to this.