We present updated cosmological constraints for the KiDS+VIKING-450 cosmic shear data set (KV450) estimated through redshift distributions and photometric samples defined using self-organising maps (SOMs). Our fiducial analysis finds marginal posterior constraints of S₈ ≡ σ₈ Ω_m/0.3 = 0.716_−0.038^+0.043, which are smaller but otherwise consistent with previous works that have applied this data set (|ΔS₈| = 0.023). We analysed additional samples and redshift distributions set up in three ways: (1) by excluding certain spectroscopic surveys during redshift calibration; (2) by excluding lower-confidence spectroscopic redshifts in redshift calibration; and (3) by considering only those photometric sources which are jointly calibrated by at least three spectroscopic surveys. In all cases, the method utilised here has been proven to be robust: we find a maximal deviation from our fiducial analysis of |ΔS₈| ≤ 0.011 for all samples defined and analysed using our SOM. To demonstrate the reduction in systematic biases found within our analysis, we highlight our results when performing redshift calibration without the DEEP2 spectroscopic data set. In this case, we find marginal posterior constraints of S₈ = 0.707_−0.042^+0.046; this is a difference, with respect to the fiducial, that is both significantly smaller and in the opposite direction with regard to the equivalent shift from previous works. These results suggest that our improved cosmological parameter estimates are not sensitive to pathological misrepresentations of photometric sources by the spectroscopy used for direct redshift calibration and, therefore, that this systematic effect cannot be responsible for the observed difference between S₈ estimates made with KV450 and Planck CMB probes.