In this letter we argue that the commonly adopted CPL parameterisation for the dark energy equation of state (EoS), is biased towards preferring such model over the constant w while the latter bounds are still compatible with their CDM values. For that we compare constraints on the EoS parameters w0 and early type wa (CPL) against those with a late time parameterisation on wa (hereafter LZ) and the constant w model, using CMB, Supernovae and BAO from DESI datasets. We found, the same as was the case with CPL model, preference for dynamical dark energy within the LZ model, but for values almost symmetrically distributed with respect to their CDM limits. This is due to the fact that the presence of w0 allows to recast each parametrisation into making it compensate the preference for w -1 in the opposite direction. To further test our hypothesis, we fixed w0 to -1 and followed a minimal investigation approach by considering models that deviates only by one free parameter following the same previous parameterisations, making them now genuinely different from each others. On the other hand, we extend our models to more general ones by, either incorporating parameterisations that group both late and early wa effects, which we name GEoS model, or by allowing the presence of two dark energy components, fluid alike DE and constant alike CC. We also complement conclusions drawn from the significance levels from the resulting confidence contours with that of the Bayes factor calculation. We found that all the variants, except the original CPL or LZ are still compatible with CDM, with likelihoods peaking close to w0 = -1, wa = 0, or 0.68 for CC, with the constant w and the late time wa (minimal LZ) being the ones with the tightest constraints. And although we found that the evidence from CPL or LZ are stronger than those for the more minimal cases, however, the preference increases further for the more generalized parameterisations, instead of penalizing the addition of the extra degrees of freedom, while still staying compatible with CDM in terms of the significance levels. We conclude that considering CPL model is not sufficient on its own, to test deviations from the simple cosmological constant model, and that it is necessary to conduct further minimal or more general approaches, when confronting models to observations, so that to localize biases, if they exist, or to better understand the outcomes from model testing and inference methods.