Experimental realizations of a one-dimensional (1D) interface always exhibit a finite microscopic width and/or a disorder correlation length. We study analytically and numerically the geometrical fluctuations of a static 1D interface with a short-range elasticity, submitted to a quenched random-bond Gaussian disorder of finite disorder correlation length, and at finite temperature. Using the exact mapping from the static 1D interface to the 1+1 Directed Polymer (DP) growing in a continuous space, we predict the existence of two temperature regimes : the influence of the disorder correlation length is erased by the thermal fluctuations at sufficiently high temperature, but turns out to be a crucial ingredient for the description of the interface fluctuations below a characteristic temperature Tc>0. Finally, we discuss the consequences of the low-temperature regime for two prototypal experimental realizations of 1D interfaces: ferromagnetic domain walls in ultrathin Pt/Co/Pt films, and interfaces in nematic liquid crystals.