The YD-condition is not equivalent to $\Psi$ being a braiding. The condition (1) is related to $\Psi$ being a morphism of $H$-modules (see the more detailed answer to your other question Reference request: compatibility conditions of four versions of Yetter-Drinfeld modules).
The Quantum Yang-Baxter equation is a consequence of the action and the coaction conditions (and the YD-condition), which are of course part of the YD-structure. These give the two hexagon axioms of a braiding (see standard references on braided monoidal categories for these conditions). $$\Psi_{V,W\otimes Z}=(id_W\otimes \Psi_{V,Z})(\Psi_{V,W}\otimes id_Z),$$ $$\Psi_{V\otimes W, Z}=(\Psi_{V,Z}\otimes id_W)(id_V\otimes \Psi_{W,Z}).$$ However, the two hexagon axioms are stronger than the quantum Yang-Baxter equation $$\Psi_{12}\Psi_{23}\Psi_{12}=\Psi_{23}\Psi_{12}\Psi_{23}.$$ This is proved by observing that $\Psi_{12}\Psi_{23}=\Psi_{V,V\otimes V}.$ This allows us to rewrite the right right hand side as $\Psi_{V,V\otimes V}(id_V\otimes \Psi_{V,V})$, but by naturality of the braiding applied to $\Psi_{V,V}$ (which is a morphism of $H$-modules by the YD-condition (1) and hence naturality applies) this equals $( \Psi_{V,V}\otimes id_V)\Psi_{V,V\otimes V}$ which can be translated into the left hand side by the same reasoning.
