The weak-temperature gradient (WTG) approximation has been a popular method used to couple convection in limited-area domain simulations to the large-scale dynamics. Two major implementations that use the WTG approximation have gained popular use over the past two decades - the Temperature Gradient Relaxation implementation and the Damped Gravity Wave implementation. Our comparison of these different WTG implementations in an idealised framework result in different model behaviour, with implications on the nature of convective self-aggregation in similarly idealised setups. A further investigation shows that the different model behaviour is caused by the treatment of the different baroclinic modes by the different WTG implementations. More specifically, we hypothesise that the ratio of the strengths of the baroclinic modes is important in determining if multiple-equilibria states are obtained under different WTG implementations. By varying the strengths of these two baroclinic modes, we are thus able to understand the differences between the WTG schemes.