Context. Supermassive black holes with masses of 10 5 -10 9 M are believed to inhabit most, if not all, nuclear regions of galaxies, and both observational evidence and theoretical models suggest a scenario where galaxy and black hole evolution are tightly related. Luminous AGNs are usually selected by their non-stellar colours or their X-ray emission. Colour selection cannot be used to select low-luminosity AGNs, since their emission is dominated by the host galaxy. Objects with low X-ray to optical ratio escape even the deepest X-ray surveys performed so far. In a previous study we presented a sample of candidates selected through optical variability in the Chandra Deep Field South, where repeated optical observations were performed in the framework of the STRESS supernova survey. Aims. The analysis is devoted to breaking down the sample in AGNs, starburst galaxies, and low-ionisation narrow-emission line objects, to providing new information about the possible dependence of the emission mechanisms on nuclear luminosity and blackhole mass, and eventually studying the evolution in cosmic time of the different populations. Methods. We obtained new optical spectroscopy for a sample of variability selected candidates with the ESO NTT telescope. We analysed the new spectra, together with those existing in the literature and studied the distribution of the objects in U − B and B − V colours, optical and X-ray luminosity, and variability amplitude. Results. A large fraction (17/27) of the observed candidates are broad-line luminous AGNs, confirming the efficiency of variability in detecting quasars. We detect: i) extended objects which would have escaped the colour selection and ii) objects of very low X-ray to optical ratio, in a few cases without any X-ray detection at all. Several objects resulted to be narrow-emission line galaxies where variability indicates nuclear activity, while no emission lines were detected in others. Some of these galaxies have variability and X-ray to optical ratio close to active galactic nuclei, while others have much lower variability and X-ray to optical ratio. This result can be explained by the dilution of the nuclear light due to the host galaxy.Conclusions. Our results demonstrate the effectiveness of supernova search programmes to detect large samples of low-luminosity AGNs. A sizable fraction of the AGN in our variability sample had escaped X-ray detection (5/47) and/or colour selection (9/48). Spectroscopic follow-up to fainter flux limits is strongly encouraged.