2 versions of the koyck distributed lag
As I explained in the edited question, the comment-question made by @caverac made me write the models out in their respective long form versions. Doing this made it obvious that an MA(1) structure can be imposed and the model is shown below.  As far as I can tell, the long form of this model cannot be transformed to a lagged dependent variable model (what I refer to as the short form ) nicely like can be in the  AR(1) and pure noise cases.

Long Form of Koyck Distributed Lag Model With MA(1) error term:

$y_{t} = \beta \times \sum_{i=0}^{\infty} \rho^{i} x_{t-i} + \epsilon_t + \rho \times \epsilon_{t-1}$