The ground state and thermodynamics of a generalized spin-1/2 Ising-Heisenberg diamond chain with the second-neighbor interaction between nodal spins are calculated exactly using the mapping method based on the decoration-iteration transformation. Rigorous results for the magnetization, susceptibility, and heat capacity are investigated in dependence on temperature and magnetic field for the frustrated diamond spin chain with the antiferromagnetic Ising and Heisenberg interactions. It is demonstrated that the second-neighbor interaction between nodal spins gives rise to a greater diversity of low-temperature magnetization curves, which may include an intermediate plateau at two-third of the saturation magnetization related to the classical ferrimagnetic (up-up-up-down-up-up-...) ground state with translationally broken symmetry besides an intermediate one-third magnetization plateau reflecting the translationally invariant quantum ferrimagnetic (monomer-dimer) spin arrangement.
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