What is a DNA clamp exactly?
Your partly right both ways.  In a sense, the DNA clamp is a protein, in another sense, it's only part of a protein.  What it actually is is what we call a protein sub-unit, which is a protein, but which binds with other protein sub-units to form complex proteins.

In order to understand this, you have to remember what a protein is.  A protein is, in essence, a chain of amino acids.  Those can be short chains, long chains, medium chains: but they're all proteins.  Sometimes, a few short proteins (sub-units) will bind together to form one long protein (protein complex).  This is what happens in the case of a DNA clamp.

Basically, the DNA clamp is (an) independent protein unit(s) which has (have) the ability to encircle the DNA strand and travel along it.  They bind to the polymerase, which is why you don't see them on your diagram.  Their function is, very simply, to make it possible for the polymerase to stay closely attached to the DNA stand.



Sources:


DNA Repair (Amst). 2009 May 1; 8(5): 570–578