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We consider the celebrated Kuramoto model with nearest neighbour interactions, arranged on a two-dimensional square lattice in presence of two kinds of noise - annealed and quenched. We focus on both the steady state and relaxation dynamics of the model. The bare model with annealed noise on finite $2D$ lattice, in the stationary state, exhibits a crossover from synchronization to desynchronization as noise strength varies. Finite-size scaling (FSS) analysis reveals that, in the thermodynamic limit, this crossover becomes a true phase transition, which is Kosterlitz-Thouless ($KT$)-type analogous to that of $2D$ $XY$ model. On the other hand, when the noise is quenched, it does not show any kind of synchronization-desynchronization phase transition in the thermodynamic limit. But we do observe a crossover from low noise-strength synchronization to high noise-strength desynchronization in finite lattices. We analyze the crossover phenomena through the linear stability of the stationary state solutions and obtain the crossover noise-strength from the onset of local instability of the unsynchronized one. The relaxation dynamics also differs for these two types of noise. In case of annealed noise, the system, in the critically ordered phase, exhibits algebraic relaxation which is described by the phenomenological Edwards-Wilkinson (EW) model of growing surface, yielding the same dynamic exponent $z =2$. In disordered phase, the system shows an exponential decay. On the contrary, the system with quenched noise, as opposed to the annealed one, always relaxes to the stationary state exponentially. Both the system-size and noise-strength dependency of the average relaxation time in the synchronized regime are also investigated.