In recent years, the Trojan Horse Method (THM) has been used to investigate the low-energy cross sections of proton-induced reactions on A = 17 and A = 18 oxygen isotopes, overcoming extrapolation procedures and enhancement effects due to electron screening. In particular, the strengths of the 20 keV and 65 keV resonances in the 18O(p, α)15N and 17O(p, α)14N reactions, respectively, have been extracted, as well as the contribution of the tail of the broad 656 keV resonance in the 18O(p, α)15N reaction inside the Gamow window. The strength of the 65 keV resonance in the 17O(p, α)14N reaction, measured by means of the THM, has been used to renormalize the corresponding resonance strength in the 17O + p radiative capture channel. As a result, more accurate reaction rates for the 18O(p, α)15N, 17O(p, α)14N, and 17O(p, γ)18F processes have been deduced, devoid of systematic errors due to extrapolation or the electron screening effect. Such rates have been introduced into state-of-the-art red giant branch and asymptotic giant branch (AGB) models for proton-capture nucleosynthesis coupled with extra-mixing episodes. The predicted abundances have been compared with isotopic compositions provided by geochemical analysis of presolar grains. As a result, an improved agreement is found between the models and the isotopic mix of oxide grains of AGB origins, whose composition is the signature of low-temperature proton-capture nucleosynthesis. The low 14N/15N found in SiC grains cannot be explained by the revised nuclear reaction rates and remains a serious problem that has not been satisfactorily addressed.