Modelling is not about selecting a priori a specific type of equation BUT rather extracting the model specifics from the data in an iterative manner as presented here https://autobox.com/pdfs/ARIMA%20FLOW%20CHART.pdf in order to optimally/opportunistically combine linear, exponential smoothing and arima components while dealing with latent deterministic structure such as pulses , level/step shifts,local time trends and/or seasonal pulses.
In your first example the deterministic structure required is a level shift (intercept change) and a few pulses with an arima (1,0,0) nearly (0,1,0) while the second example it is simply two pulses with an arima (0,1,0) .
first example:
There is a very clear pattern in the data as shown here 
. Your 20 values are adequately described by a hybrid model using an AR(1) and a step/level shift along with 3 pulses . 
and here 
and here 
The tools (approaches) that you were considering are presumptive in form whereas your data has it's own message. This data has not only a strong memory but has been affected by external activity causing the step.level shift and the 3 pulses.
here are the forecasts for the next 13 periods 
The method used here to form the model is called Intervention Detection as detailed here and everywhere else http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html . Search SE for "INTERVENTION DETECTION" . It might behoove you to investigate the true cause of the level/step shift in order to more intelligently forecast this series.
Here is the Actual and Cleansed plot 
The reason that arima (memory) doesn't work (alone) is that there is determinstic structure in the data .
second example:
This is also a hybrid model arima (1,1,0) with two pulses reflecting external deterministic inputs. The Actual/Fit and Forecast is here 
with equation here 
and here 
with statistical summary here 
and for
ecasts here . The Actual and cleansed graph is here 