Fake Books / Busker Books - the same thing.
The originated in America and were for many years illegal. I have one from the second world war, issued to services musicians. Though spiral bound, 1000 tunes and some with lyrics, this book has been printed from hand-written scores - single melody line and chord symbols.
The USA term "Fake" does indeed refer to "faking it" or improvising when it comes to the harmonies. The chord symbols were used for accordions, guitar, piano, organ - any solo instrument that has a chord orientated system.
IMP publish a whole series of them but in the 60's, every major publisher produced their own, containing popular tunes from their specific catalogues.........Chappell, Feldman, Lawrence Wright, Campbell Connelly......we have a collection of around 50 at home.
There is also a series of American "Fake" books - more recent, that gives you alternative harmonies - and pretty stunning they are to. Some of the chord symbols look like quadratic equations. HOWEVER, it is not the chord itself, but how the chord is arranged under the melody line. It takes a fair bit of studying to achieve what the arranger possibly had in mind (or ear!).
Some of the books I have show not only the original composers harmonic line, but show the re-harmonised line as well - it makes interesting comparison.
These are "Altered Chord" Fake Books.
Worth looking out for on Ebay !
Any more info? - pm me!