Sanhedrin 30a

What is the legal distinction between "Eidim" and "Beis Din"?  They are two distinct legal categories.  I have said many times that Halacha sees Eidim as being more than just "witnesses". Eidim are in a way "Officers of the Court", acting together as  "Notaries Public" who validate proceedings with their testimony.  The Torah uses the expression "Two or three eidim", so there is such a thing as a triad of eidim.

But a Beis-Din, the Court itself, has a power above and beyond that of the Legal Force of the Eidim.  What is the nature of this difference?

Why is a statement made in front of people acting as witnesses more impeachable than a statement made in front of a Beis Din.

Why does the number three mean so much more than the number two? To strengthen the point, I will point out that the Gemara itself presents a view that a Beis Din of two has standing according to Shmuel.  Yet there is seemingly still an advantage in a Beis Din of two members (Beis Din Chatzuf) over an eidim team of three!  Why?