The quantum kagome antiferromagnets YCu₃(OH)₆O_xCl_3−x (x = 0,1/3) are produced using a unified solid state synthesis route for polycrystalline samples. From structural refinements based on neutron diffraction data, we clarify the structure of the Y₃Cu₉(OH)₁₈OCl₈ (x = 1/3) compound and provide a revised chemical formula. We use muon spin relaxation, as a local probe of magnetism, to investigate the exotic low-temperature properties in the two compounds. In agreement with the low-temperature neutron diffraction data, we find no evidence for long range ordering in both materials but they exhibit distinct ground states: while disordered static magnetism develops in the x = 0 compound, we conclude on the stabilization of a quantum spin liquid in the x = 1/3 one, since the local fields remain fully dynamical. Our findings are in contrast to previous reports based on thermodynamical measurements only. We then discuss their origin on the basis of structural details and specific heat measurements. In particular, the x = 1/3 compound appears to realize an original spatially anisotropic kagome model.