Phenomenological parameterization of the perturbed Einstein’s equations have become an important route to testing General Relativity (GR) at cosmological scales. This is often done by constraining a pair of modified gravity (MG) parameters. We explore here constraints on one MG parameter at a time while fixing the other at its GR value. This allows one to analyze various models while benefiting from a stronger constraining power from the data. We find cases with stronger tension with GR than two-MG-parameter approaches. For example, models with (μ=1, η) and (μ, η=1) exhibit a 3.86-σ and 3.77-σ departure from their GR values when using Planck18+SNe+BAO data. We find no tension with GR for models with the MG parameter Σ fixed to its GR unity value. Using a Bayesian model selection analysis, we find one-parameter MG models that are moderately favored over ΛCDM when using all dataset combinations except Planck CMB Lensing and DES data. Namely, Planck shows a moderate tension with GR that only increases when adding any combination of RSD, SNe, or BAO. However, adding Planck CMB Lensing or DES data respectively diminishes or removes these tensions, which can be attributed to the ability of lensing in constraining the MG parameter Σ. The two overall groups of datasets are found to have a dichotomy when performing consistency tests with GR, which may be due to systematic effects, lack of constraining power, or modelling. These findings warrant further investigation using more precise data from ongoing and future surveys.