What are really asking is "What is the taxonomy of a time series model ?" and probably should be the first point of order when teaching/learning time series analysis. Kudos to you !
Time series forecasts can include
1) User specified predictor series which might have a contemporaneous and/or lag effects. These predictors can be either stochastic or deterministic .
2) Latent deterministic structure such as level shifts , trend changes , seasonal pulses and/or pulses . Note the plural suggesting generality of approach and not presumption.
3) Memory effects of which ARIMA is the most general form being a generalization of simple weighted averages like exponential smoothing (the Brown model ) and other simple procedures like a K period moving average where K is assumed and the k weights are specified either implicitly or explicitly.
Your two particular examples can be characterized as follows :
Example 1 is a particular deterministic effect (type 1) with 1 trend based upon two data points
Example 2 is a type 3 model where it is assumed that the adjusting for previous values is all that is needed AND there are no latent deterministic structures/features in the data.
Modern approaches ( read general approaches ) consider a hybrid model integrating/optimizing one or more of the three types that I have detailed here.
I suggest that you follow this very broad BUT good question with a more detailed one and actually present data be it textbook data , real data or simulated data and let some of the responders actually illustrate how to build a model in an educational step-by-step approach. Alternative solutions can be rendered with commentary regarding pro's and con's.
All three of these possible components to the time series model must be tested for constancy of the model error variance over time AND constancy of the model parameters over time ...fulfilling the stationarity/reproducability requirements for the model errors .
A visual of the three components ("X" being user-specified ; "I" being latent and waiting to be discovered and the error process ("e") reflecting the ARIMA/memory component or the currently unknown components is here http://www.autobox.com/pdfs/SARMAX.pdf.
Useful models are not necessarily simple models just complicated enough to characterize the data in a minimally sufficient form where all estimated coefficients are necessary .
Each time series has it's own distinct features that require identification in order to separate signal and noise.