I have occasionally seen the plane to which you are referring referred to as just that - centre line.
But more usually, it is referred to as 'amidships'. Usually on the profile, less so on the half-breadth plan. And the symbol showing it is of a circle intersected at its own centre with the two halves of another circle 'back-to-back'.
As the line to which you refer is not a line of symmetry (where the for-and-aft one is, of course and is subsequently very important in the building shed too) it is of more importance in calculations, where it is called the centre of gravity.
Though again, more usefully on the profile, than on the half-breadth plan. For performing trim and sailplan calculations and geometry for example.
In the building shed it is most usually simply referred to as the number of the station on the loft plan as has the greatest beam. So: 'Station No 10', for example. Which is important in that this is also the position where the molded line from the loft plan is transferred to the opposite face of the moulds or frames, to allow for the change in direction of their bevel. (The lofted face is always the aft face on the fwd frames or moulds and the fwd face on the aft ones).
... As set around the line you refer to, basically.
But there is nothing says that the face of any frame nor, for that matter, the point of greatest beam of any part of the hull - frame or skin - will correspond with the centre of gravity of the boat.
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