Designs for discriminating between two linear regression models are studied under $Lambda$-type optimalities maximizing the measure for the lack of fit for the designs with fixed model inadequacy. The problem of selecting an appropriate $Lambda$-type optimalities is shown to be closely related to the estimation method. $Lambda$-type optimalities for the least squares and minimum bias estimation methods are considered. The minimum bias designs are suggested for the designs invariant with respect to the two estimation methods. First order minimum bias designs optimal under $Lambda$-type optimalities are then derived. Finally for the case where the lack of fit test is significant, an approach to the construction of a second order design accommodating the optimal first order minimum bias design is illustrated.