What is the difference between normal relative clauses and the ones with という added?


Let's compare these 2 sentences  

  1. 私が来た意味
  2. 私が来た **という** 意味   

I think both mean the same, but my intuition (although it may be wrong) tells
me that 私が来た意味 more leans to the meaning of **The reason why i came** , while
the latter more leans to **It means "i came"** as in explaining what does 私が来た
mean.  
But i feel like both can mean the same in certain context.  

Am i wrong?  
If yes, is there any difference, not only for the sentences above, but just in
general?  
If there is, what causes it?

