Definitive maps (and accompanying statements) will only show the legal existence of footpaths (FP), bridleways (BW), the relatively new status of restricted byway (RB) and byways open to all traffic (BOATS). It came about via legislation in 1949, and only shows those that were originally claimed by parishes in the early 1950's from historical evidence, and subsequent legal changes since that time - diversions, creations and extinguishments.
Having worked with def maps on a day to day basis for close on 20 years, I can vouch that the OS depict the PRoW network very well indeed. Obviously the scales used (25K and 50K) do limit accuracy in representing the 'exact line' but for most countryside users purpose they are more than sufficient.
The def map does not show 'white roads' as there is not such legal designation or definition. 'White Road' is just a descriptive term for a map feature, since that's how they are depicted, much the same as green squares, blue squiggly lines and brown splodges. On the ground they get referred to as 'green lanes', again this is just a descriptive 'catch all' term that has no legal standing. The OS use 'twin parallel lines with white infill' (aka white roads) to depict a range of tracks from urban residential roads to private farm access tracks, so coming out with a sweeping statement such as "The majority of white roads shown on the OS maps in England are in fact public roads" is utter tosh.
UUCRS (as I stated above) have public access rights, although only passage on foot can certain and any higher rights are in many cases questionable. However you'll find these on the local highway authority's (the county council usually) 'List of Streets'... not the definitive map.
Anyhoo, here's the IMBA take on this -
http://imba.org.uk/what-are-white-roads/