yt=δ+yt−1+ϵt is the leftmost plot while yt=β0+β1t+ϵt is the rightmost plot. Both original time series have very similar acf's and pacf's thus model identification is difficult as the two competing models can't be distinguished from each other simply using the observed data.
However if one incorrectly specifies yt=β0+β1t+ϵt for the leftmost series the residuals will not be free of structure suggesting misidentification. Similarly if you specify yt=δ+yt−1+ϵt for the rightmost data set you will also get a residual series that is not free of structure.
Discerning between a stochastic/adaptive model and a fixed/deterministic model can get a little bit trickier when you have break points in the data and possible level/step shifts with possible error variance or model parameter transience.
Model identification strategies need to consider the potential of a hybrid model where both stochastic and/or deterministic structure are possibly present in the data as described here in an iterative process https://autobox.com/pdfs/ARIMA%20FLOW%20CHART.pdf.
Ultimately it is the analysis of model residuals that is the final arbiter of which approach is more correct for any given data set.