The problem appeared after the multi-exposure mode was used on the camera (Mark III). At this moment only the silent shooting modes are applied in the Live View mode (i.e. I don't get the "clack clack clack" of the mirror in between shots as it was before; the menu choices (silent/regular) does not affect the shooting modes used by the camera). When the shooting via the viewfinder is applied I can choose between the silent (s) and regular shooting modes, and the camera shoots in accordance with the choosing mode. Thanks.
If you were getting mirror movement when using Live view, it is because you had selected the Quick Mode AF which uses the camera's Phase detection AF (normally used when shooting with the viewfinder) instead of the Live Mode or Face detection Live Mode which use the camera's sensor based Contrast Detection AF.
For instructions in how to select a focus mode in Live View, see page 209 of your EOS 5D Mark III Instruction Manual. For how to shoot using the Quick Mode, see page 214.