B,
I guess that you're a new chum .. be prepared for the jargon to confuse you for quite some time until you get your head around it ..
The problem is written into the ships log book
The reference is to the aircraft's (technical) log (book). This is a document which can take many formats and names but, basically, is a vehicle to record technical data and unserviceabilities.
MCOs
This practice depends on the State's rules. Some places don't allow such things, others do. An alternative expression used in the Australian military is CFU (carried forward unserviceability). Generally the civil MEL provides a similar process in the end play although the procedural steps might vary.
If there is a problem (engine, structural etc,) where does the pilot log it?
There is a multiplicity of "logs" or logbooks.
The airworthiness records include airframe, engine, propeller, etc., logbooks which, in general, aren't used routinely by flightcrew, especially in heavy aircraft. These documents provide the maintenance history in considerable detail and are the normal province of the maintenance folk.
The usual interface between flight and ground crew will be via the technical log (by whatever name). This should be viewed as being a subsidiary component of the maintenance logbooks referred to in the previous paragraph. If you like one could view the tech log as being a very robust, systematised method of writing notes between the two groups. In addition, it provides a convenient vehicle for flightcrews to record other data of use to maintenance control functions.
In general we try to keep all records OFF the aircraft ... consider what happens if the aircraft has an accident .... we end up with no records to assist the investigation. In some States, and for some circumstances, even the flight manual is left off the aircraft.