Question
according to the scheme for sedimentary rock identification, which characteristic determines whether a rock is classified as a shale, a siltstone, a
sanostone, or a conglomerate?

According to the scheme for Sedimentary Rock Identification, the particle size of the sediments within the rock characteristic determines whether a rock is classified as a shale, a siltstone, a  sanostone, or a conglomerate.

The given four types of rocks i.e. conglomerate, siltstone, sandstone and shale are the classic type of Sedimentary rocks. These rocks have different particle sizes of sediments present in them, which gives a particular characteristic to their structure.  

Conglomerate is the most coarse grained rock with size of particles 64 mm to >256 mm, and having rounded grains.  Breccia or siltstone is finer, with particle size from 2 mm to 64 mm, and angular grains.  Sandstone are much finer with grains ranging in size from 2 mm to 1/16 mm.  Shale are having the finest grains ranging in size less than 1/16 mm.