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We present updated cosmological constraints for the KiDS+VIKING-450 cosmic shear data set (KV450), estimated using redshift distributions and photometric samples defined using self-organising maps (SOMs). Our fiducial analysis finds marginal posterior constraints of $S_8\equivσ_8\sqrt{Ω_{\rm m}/0.3}=0.716^{+0.043}_{-0.038}$; smaller than, but otherwise consistent with, previous work using this data set ($|ΔS_8| = 0.023$). We analyse additional samples and redshift distributions constructed in three ways: excluding certain spectroscopic surveys during redshift calibration, excluding lower-confidence spectroscopic redshifts in redshift calibration, and considering only photometric sources which are jointly calibrated by at least three spectroscopic surveys. In all cases, the method utilised here proves robust: we find a maximal deviation from our fiducial analysis of $|ΔS_8| \leq 0.011$ 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 $S_8=0.707_{-0.042}^{+0.046}$; a difference with respect to the fiducial that is both significantly smaller than, and in the opposite direction to, the equivalent shift from previous work. These results suggest that our improved cosmological parameter estimates are insensitive to pathological misrepresentation 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 $S_8$ estimates made with KV450 and Planck CMB probes.