I think the "random-effects" setting is a special case of the "random-design" setting. For instance suppose the Z matrix coincides with the X matrix in the "random-effects" setting and assume it is the normal mixed linear model. In that case a typical component in the "random-effects" setting looks like

x(fixed-parameter + N(0,var)) = N(x*fixed-parameter, x * var * x^t).

On the other hand in the normal "random-design" setting, a typical component looks like

fixed-parameter(x + N(0, var)) = N(x*fixed-parameter, fixed-parameter * var * fixed-parameter^t).

In both settings, we want to estimate "fixed-parameter" and the covariance matrix "var". With a change of variables for the covariance matrix, we can jump between the two settings.