What's the definition of "Dynamic Regression Models"? I am trying to learn about Dynamic Regression models. However, the sources on the topic is (relatively) few compared to other TS topics, and so I cannot really get a grasp of where to start. I really want to get a specific definition of what a dynamic regression model is. What for example is the difference between an ARIMA model with external regressors and a dynamic regression models?
Is it simply a AR distributed lag model with ARMA errors?
 A: No problem but I'd definitely try to get  my hands on some kind of textbook or lecture notes. The following  is not a formal definition but here goes my attempt: To me, a DRM is any model which is time dependent, in the sense that, the next value of an observation (on the right hand side of the model ), at the next point in time, changes the model forecast. So, suppose you are sitting at time $t$, and you have a model say 
$y_{t+1} = \alpha y_{t} + \beta_{1} x_{t} + \beta_2 x_{t-1} + \epsilon_{t+1}$. 
Now, depending on the convention,  this model would probably be referred to as an ARDL(1,1) which is a specific case of a class of dynamic regression models called auto-regressive distributed lag models. Note that, when sitting at time $t$, this model ( after some algebraic manipulation and assumptions about $x_{t}$  and $\epsilon_{t+1}$ ) can provide  a forecast for say $y_{t+h}$ where $h$ is the forecast horizon. My point here is that that $h$ does not necessarily need to be assumed to be one.  Next , suppose one unit of time passes so that a new observation, $x_{t+1}$ arrives at time $t+1$. This means that the new model forecast of $y_{t+h+1}$ will get updated to reflect this new observation. Note also (and this can be crucial) , the error terms can play a big role also ( be say MA(1) or AR(1)  rather than just IID-normal ) so, even if the new observation of $x$ at time $t+1$ was the same as the observations at time $t$ and time $t-1$, and there was no lagged dependent variable, $y_{t-1}$ like there is above, the forecast at time $t+h+1$ could still change !!!!!! This is a powerful concept and would not be the case in a more "static" type of regression model. Essentially, this is what is meant by the term dynamic, namely that, because of the lagged dependent variable on the RHS and or the error terms, the new $x_t$ observations that arrive in the future could stay constant for many periods and the model forecasts could still change.
I hope this helped a little. It  isn't really an answer but I put it in the answer section because there's more space. I happen to have a lot of experience with dynamic regression models so I've collected many lecture notes and texts over the years. When I get a chance this weekend, I'll try to send some useful links but the dynamic concept is pretty much as simple as what I explained. There's nothing terribly complex regarding the term "dynamic" but it can sound intimidating.  Estimation of parameters can be a problem sometimes because of multiple parameters, multi-collinearity etc. Stability of parameters over time is another possibly problematic issue. Remind me if I forget to post links.
Here are some links.
https://www.reed.edu/economics/parker/312/tschapters/S13_Ch_3.pdf
http://www-personal.umich.edu/~franzese/DeBoefKeele.2008.TakingTimeSeriously.pdf
https://www.nuff.ox.ac.uk/politics/papers/2005/Keele%20DeBoef%20ECM%20041213.pdf
https://pdfs.semanticscholar.org/presentation/fd49/2458fbe607f3bf19ff28aa872b4980ebd629.pdf
https://jdemeritt.weebly.com/uploads/2/2/7/7/22771764/timeseries.pdf
http://web.thu.edu.tw/wichuang/www/Financial%20Econometrics/Lectures/CHAPTER%2015.pdf
The literature is beyond huge so above is only touching it. The reason it is so vast because these models arise in statistical time series, econometric time series and DSP but all using different notation and sometimes different terminology. 
Most of the links above, if not all, have  an econometric-statistical time series connection. Also, keep in mind that,  although some of the above links will touch on the koyck distributed lag (1954), that model has a whole literature unto itself. Definitely, DRM's are a deceptive area because there's a TON to know about a relatively straightforward time-series topic.I I think that my original statement above about them being straightforward could be misleading because their flexibility makes them quite powerful and provides a lot to discuss and learn about them. All the best.
Oh, one last thing:  you may want to check out economics.stackexchange.com also. It's obviously more economics but econometric time series is discussed once in a while and there are some really good people over there just like there are here. 
