Gong,
In answer to your question I doubt any manufacturer nowadays uses chlorate/red phosphorus (even pacified with boron carbide) in its primers; but it was used in the past. Its generally considered too sensitive, hygroscopic and its decomposition products are corrosive to steel barrels... Not ideal.
The materials used in modern primers vary from manufacturer to manufacter, but broadly fall into 9 categories. The most common composition type (as 408 said) is what is called SINOXID which has a number of common derivatives. These normally contains Lead Styphnate, barium nitrate and antimony suphide, although the antimony sulphide is occasionally replaced by calcium silicide or aluminium and lead styphnate may be replaced or augmented by tetrazene or diazodinitrophenol.
SINTOX is another alternative normally used in lead free primers and is generally composed of diazodinitrophenol or tetrazene, zinc peroxide and titanium, although derivatives containing strontium compounds are apparently not uncommon in the US.
Rimfire cartridges are pretty much invariably Lead Styphnate, Barium nitrate compositions, but some include phosphorus compounds (at least according to SEM/EDX) and almost all use glass as a frictionator...
Hope this helps
R