The in-house part numbering scheme allows Engineering to see what is already commonly used, Purchasing can get the best pricing for higher volumes - but it's another parts system to maintain. Vendor qualification, parts qualification, and designs can end up with the same or similar parts from different manufacturers which can cause surprises.
If you're using in-house part numbers, stick to that in the schematics.
Never put mfgr specifics in the schematics then, as you are bastardizing your in-house part numbering system.
You will always have multiple vendors that can supply a part, some with subtle differences. Someone, usually engineering, decides what parts are suitably qualified for the in-house number. Exceptions or special parts should have a unique in-house part number, not a note on the schematic to use X part for X manufacturer.
Otherwise, only use manufacturer's part numbers in your schematics.
It depends on your companies size and if the designs are really sensitive to a component's specifics.