In this work, we expand on the XENON1T nuclear recoil searches to study the individual signals of dark matter interactions from operators up to dimension eight in a chiral effective field theory (ChEFT) and a model of inelastic dark matter (iDM). We analyze data from two science runs of the XENON1T detector totaling 1 t×yr exposure. For these analyses, we extended the region of interest from [4.9,40.9] keVNR to [4.9,54.4] keVNR to enhance our sensitivity for signals that peak at nonzero energies. We show that the data are consistent with the background-only hypothesis, with a small background overfluctuation observed peaking between 20 and 50 keVNR, resulting in a maximum local discovery significance of 1.7⁢𝜎 for the Vector⊗Vectorstrange ChEFT channel for a dark matter particle of 70 GeV/𝑐2 and 1.8⁢𝜎 for an iDM particle of 50 GeV/𝑐2 with a mass splitting of 100 keV/𝑐2. For each model, we report 90% confidence level upper limits. We also report upper limits on three benchmark models of dark matter interaction using ChEFT where we investigate the effect of isospin-breaking interactions. We observe rate-driven cancellations in regions of the isospin-breaking couplings, leading to up to 6 orders of magnitude weaker upper limits with respect to the isospin-conserving case.