This thread may be getting unbearably tedious to some but allow me (try and stop me!) to quote some more from THE OXFORD COMPANION TO SHIPS AND THE SEA with regard to "fake" which they define as "a complete turn of a rope when it has been coiled either on deck or on a drum. When a rope has been properly faked down, it is clear for running, each fake running out without fouling those below it." "Flake" was earlier defined as coils laid out for inspection rather than running.

Linguists might argue that they are the same term, with the "l" having worn away over the last hundred years through hard usage by those who actually sail.;-)