Six persons lived in a castle wherein each occupant had his own apartment. One morning, two of the occupants were dead:

A died because he sat on a poisoned needle that was hidden in his bed. In the last moment of his life, he scratched into the bedsheet with the needle that he took out of his backside: M = C
It was understood to mean:

A: "My murderer is C."

B was suffocated by a tampered Baroque bed canopy that crashed down to him. In the last seconds of his life, he carved a message on the wooden bedstead with a cork-screw in the one hand hanging out:

B: "My murderer is F."

Collection and evaluation of evidence by the police proved that each victim had been murdered by one of the other occupants. C, D, E, and F refused to answer police questions. There was hope that they would speak at the trial. But since the court room was bombed near the end of the proceedings and no one survived inside, the outcome of the court case remained unknown until now.
Seventy years later, two half burnt pieces of the court record were found in a huge forgotten box full of stones and ashes. The first piece of paper contains the statements of C, D, E, and F:

C: "F is a murderer or D is lying as always."

D: "A's or B's statement is erroneously false, or C's statement is true."

E: "If what I say is true, then the murderer of A and the murderer of B are not the same."

F: "If C murdered B or E is telling the truth, then I killed A."

The second piece of paper contains a snippet of the court decision:

"...the A case and the B case are closed and confirm the rule: No murderer without a lie..."

Can you prove that there is a unique solution for one of the victims and that there are two solutions for the other one?

(The word "or" was used to mean "at least one, and maybe both".