Toën, Kapranov, and unweighted triangle counts are distinct constructions

Toën's $\mathcal{D}\mathcal{H}(\mathcal{T})$ is built from isomorphism classes of perfect modules in a dg-enhancement. Its coefficient counts homotopy-cofiber arrows and includes the alternating higher-Ext factor of Toën derived Hall numbers have the higher-Ext orbit formula. The dg enhancement is essential: an unweighted count of triangle orbits is not the same coefficient; the finite-vector-space calculation on the companion examples page illustrates the missing higher-homotopy factor.

Kapranov's lattice Hall algebra for a finitary hereditary category has generators indexed by shifts of heart objects and additional group-algebra generators $k_{\alpha }$ indexed by $\alpha \in K_0(\mathcal{A})$, the group generated by object classes modulo short-exact-sequence relations. Its lattice and $K_0$ data are part of its definition. Toën's derived Hall algebra instead has a basis indexed by perfect derived objects and uses homotopy weights. These are distinct constructions; this remark makes no claim that the resulting algebras are non-isomorphic or that one embeds in the other. Schiffmann's §5.6 displays both presentations and explicitly says they are closely related but not equivalent constructions.

Motivic Hall products use stack functions and a chosen motivic coefficient theory; cohomological Hall products use equivariant cohomology correspondences and their grading/push-pull data. Their examples and hypotheses belong to the separate pages motivic-hall-algebras-and-integration and cohomological-hall-algebras-and-quiver-correspondences. They are not consequences of the Toën product defined here.