Half-Life and Precision Shape Measurement of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mn>2</mml:mn><mml:mi>ν</mml:mi><mml:mi>β</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:math> Decay of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mmultiscripts><mml:mrow><mml:mi>Te</mml:mi></mml:mrow><mml:mprescripts/><mml:none/><mml:mrow><mml:mn>130</mml:mn></mml:mrow></mml:mmultiscripts></mml:mrow></mml:math>

We present a new measurement of the 2νββ half-life of ^{130}Te (T_{1/2}^{2ν}) using the first complete model of the CUORE data, based on 1038 kg yr of collected exposure. Thanks to optimized data selection, we achieve a factor of two improvement in precision, obtaining T_{1/2}^{2ν}=(9.32 _{-0.04}^{+0.05}stat _{-0.07}^{+0.07}syst)×10^{20} yr. The signal-to-background ratio is increased by 70% compared to our previous results, enabling the first application of the improved 2νββ formalism to ^{130}Te. Within this framework, we determine a credibility interval for the effective axial coupling in the nuclear medium as a function of nuclear matrix elements. We also extract values for the higher-order nuclear matrix element ratios: second-to-first and third-to-first. The second-to-first ratio agrees with nuclear model predictions, while the third-to-first ratio deviates from theoretical expectations. These findings provide essential tests of nuclear models and key inputs for future 0νββ searches.