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We use measurements of 59/58 quasars (QSOs), over a redshift range $0.0041\leq z \leq 1.686$, to do a comparative study of the radius--luminosity ($R-L$) and X-ray$-$UV luminosity ($L_X-L_{UV}$) relations and the implication of these relations for cosmological parameter estimation. By simultaneously determining $R-L$ or $L_X-L_{UV}$ relation parameters and cosmological parameters in six different cosmological models, we find that both $R-L$ and $L_X-L_{UV}$ relations are standardizable but provide only weak cosmological parameter constraints, with $L_X-L_{UV}$ relation data favoring larger current non-relativistic matter density parameter $Ω_{m0}$ values than $R-L$ relation data and most other available data. We derive $L_X-L_{UV}$ and $R-L$ luminosity distances for each of the sources in the six cosmological models and find that $L_X-L_{UV}$ relation luminosity distances are shorter than $R-L$ relation luminosity distances as well as standard flat $Λ$CDM model luminosity distances. This explains why $L_X-L_{UV}$ relation QSO data favor larger $Ω_{m0}$ values than do $R-L$ relation QSO data or most other cosmological measurements. While our sample size is small and only spans a small $z$ range, these results indicate that more work is needed to determine whether the $L_X-L_{UV}$ relation can be used as a cosmological probe.