R Coronae Borealis and dustless Hydrogen-deficient Carbon stars have different oxygen isotope ratios

R Coronae Borealis (RCB) and dustless Hydrogen-deficient Carbon (dLHdC) stars are believed to be remnants of low mass white dwarf mergers. These supergiant stars have peculiar hydrogen-deficient carbon-rich chemistries and stark overabundances of $^{18}$O. RCB stars undergo dust formation episodes resulting in large-amplitude photometric variations that are not seen in dLHdC stars. Recently, the sample of known dLHdC stars in the Milky Way has more than quintupled with the discovery of 27 new dLHdC stars. We present medium resolution (R$\approx3000$) near-infrared spectra of 20 newly discovered dLHdC stars. We confirm that unlike RCB stars, dLHdC stars do not show strong blueshifted ($>200$ km s$^{-1}$) He I 1.0833 $\mu$m absorption, suggesting the absence of strong, dust-driven winds around them. We also present medium resolution (R$\approx3000-8000$) $K$-band spectra for 47 RCB stars. We measure the $^{16}$O/$^{18}$O ratios of 7 dLHdC and 31 RCB stars that show $^{12}$C$^{16}$O and $^{12}$C$^{18}$O absorption bands, and present the largest sample of values of $^{16}$O/$^{18}$O for dLHdC and RCB stars to date. We find that six of the seven dLHdC stars have $^{16}$O/$^{18}$O $<0.5$, while 26 of the 31 RCB stars have $^{16}$O/$^{18}$O $>1$. We conclude that most dLHdC stars have lower $^{16}$O/$^{18}$O than most RCB stars. This confirms one of the first spectroscopic differences between RCB and dLHdC stars. Our results rule out the existing picture that RCB stars represent an evolved stage of dLHdC stars. Instead, we suggest that whether the white dwarf merger remnant is a dLHdC or RCB star depends on the mass ratios, masses and compositions of the merging white dwarfs.