Magnetic correlations in subsystems of the misfit [Ca₂CoO₃]_(0.62)[CoO₂] cobaltate

[Ca$_2$CoO$_3$]$_{0.62}$[CoO$_2$], a two dimensional misfit metallic compound, is famous for its rich phases accessed by temperature, $i.e.$ high temperature spin-state transition, metal-insulator transition (MIT) at intermediate temperature ($\sim$ 100 K) and low temperature spin density wave (SDW). It enters into SDW phase below T$_{MIT}$ which becomes long range at 27 K. Information on the independent role of misfit layers (rocksalt/Ca$_2$CoO$_3$ \& triangular/CoO$_2$) in these phases is scarce. By combining a set of complementary macroscopic (DC magnetization and resistivity) and microscopic (neutron diffraction and X-ray absorption fine structure spectroscopy) measurements on pure (CCO) and Tb substituted in the rocksalt layer of CCO (CCO1), magnetic correlations in both subsystems of this misfit compound are unraveled. CCO is found to exhibit glassiness, as well as exchange bias (EB) effects, while CCO1 does not exhibit glassiness, albeit it shows weaker EB effect. By combining local structure investigations from extended X-ray absorption fine structure (EXAFS) spectroscopy and neutron diffraction results on CCO, we confirm that the SDW arises in the CoO$_2$ layer. Our results show that the magnetocrystalline anisotropy associated with the rocksalt layer acts as a source of pinning, which is responsible for EB effect. Ferromagnetic clusters in the Ca$_2$CoO$_3$ affects SDW in CoO$_2$ and ultimately glassiness arises.