Scholl-Den start of section 3.
One can formulate a weaker conjecture by simply replacing the subscript Z motivic cohomology by an unspecified subspace.
Kim on p 13 says that the general conjecture about orders of vanishing is expressed in terms of a conjectural cat of mixed motives over Z, as dim Ext1 - dim Hom. Kim in general has a very nice structure on his overview, explaining what happens the central, near-central points etc, and explaining the rational and transcendental terms. He says the conceptual structure falls into two parts: Relations between and L-functions and Ext groups in cats of motives, and geometric interpretation of these Ext groups.
There is a formulation in Schneider.
nLab page on Beilinson conjectures SPECIAL VALUES STATEMENTS