Combining deep Space Telescope (̋ST) images and extensive data from the Multi-Unit Spectroscopic Explorer, we present new mass models of the cluster MACS J1149. 5+2223, strongly lensing the supernova (SN) Refsdal, fully exploiting the source surface-brightness distribution of the SN host for the first time. In detail, we incorporated 77, 000 ̋ST pixels, in addition to the known 106 point-like multiple images, in our modeling. We considered four different models to explore the effect of the relative weighting of the point-like multiple image positions and flux distribution of the SN host on the model optimization. When the SN host's extended image is included, we find that the statistical uncertainties of all 34 free model parameters are reduced by factors ranging from one to two orders of magnitude compared to the statistical uncertainty of the point-like only model, irrespective of the adopted different image weights. We quantified the remarkably increased level of precision with which the cluster's total mass and the predicted time delays of the SN Refsdal multiple image positions can be reconstructed. We also show the delensed image of the SN host, a spiral galaxy at z_ =1. 49, in multiple ̋ST bands. In all those applications, we obtain a significant reduction of the statistical uncertainty, which is now below the level of even the small systematic uncertainty on the mass model that could be assessed by the different approaches. These results demonstrate that with extended image models of lensing clusters it is possible to measure the cluster's total mass distribution, the values of the cosmological parameters, and the physical properties of high-redshift sources with an unparalleled precision, making the typically not-quantified systematic uncertainties now crucial. Hubble SN
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