How can I identify these time series processes? (AR/MA/ARIMA/random walk with drift) I don't understand how one would identify the stochastic process of the following models, if they are AR or MA or ARIMA etc.

Consider the following models estimated over a sample $t = 1, 2, \dots,T$. In each case $\epsilon_t$ is white noise and $p$
  and $\mu$ are less than one in absolute value.
\begin{align}
 y_{t} &=\alpha+\rho y_{t-1}+\epsilon_{t} \tag1\\ 
 y_{t} &=\alpha+y_{t-1}+\epsilon_{t} \tag2\\
 y_{t} &=\alpha+\epsilon_{t}+\mu \epsilon_{t-1}\tag3 \\ 
\Delta y_{t}&=\alpha +\rho \Delta y_{t-1}+\epsilon_{t}+\mu \epsilon_{t-1} \tag4
\end{align}

 A: Compare the models to the definitions.  Using your notation, I will paraphrase Wikipedia articles because these are accessible for further study.
All four models rely on a white noise series of random values $\epsilon_t.$
The Autoregressive (AR) model of order $p$ with parameters $\phi_1, \ldots, \phi_p,$ a "constant" $c$, and white noise $\epsilon_t$ is given by
$$y_t = c + \phi_1 y_{t-1} + \cdots + \phi_p y_{t-p} + \epsilon_t.\tag{AR}$$
"Autoregressive" means each successive value $y_t$ is modeled as a linear combination of immediately preceding values (that is, "regressed" on them).  Comparing this to the options in the question, it is apparent that (1) can be written in this form with $p=1,$ $\alpha=c$, and $\rho=\phi_1.$
The Moving Average (MA) model of order $q$ with parameters $\theta_1, \ldots, \theta_q$ and constant $c$ is given by
$$y_t = c + \epsilon_t + \theta_1\epsilon_{t-1} + \cdots + \theta_q \epsilon_{t-q}.\tag{MA}$$
The expression on the right is a weighted combination--or, through abuse of the term, a weighted "average"--of the immediately preceding white noise terms, whence the term "moving average".  This creates a correlation between $y_t$ and successive values $y_{t+1}, \ldots$ because they share some of the $\epsilon_j$ in their definitions.
It is apparent that (3) is in this form with $q=1,$ $\alpha=c,$ and $\mu=\theta_1.$
The Autoregressive Integrated Moving Average (ARIMA) model of order $(p,d,q)$ combines the AR and MA models with the idea of "differencing" a time series.  (The "Integrated" part undoes the differencing.)  It is combines an AR$(p)$ and MA$(q)$ model for the differenced time series $\Delta y_t = y_t - y_{t-1}.$  ($d$ is the number of times differencing is applied.) The combined "ARMA" model is given by $p, q,$ and coefficients $\phi_1, \ldots, \phi_p;$ $\theta_1,\ldots, \theta_q$ as
$$y_t = \left(\phi_1 y_{t-1} + \cdots + \phi_p y_{t-p}\right)\, +\, \left(\epsilon_t + \theta_1 \epsilon_{t-1} + \cdots + \theta_q \epsilon_{t-q}\right).\tag{ARMA}$$
It is apparent that (4) posits an ARMA model for the differences $\Delta y_t.$  This makes it an ARIMA$(1,1,1)$ model.
Finally, the special case where each successive value $y_t$ is obtained by displacing the preceding value $y_{t-1}$ randomly by a zero-mean value $\delta_t$ is known as a "random walk."  When this white noise term $\delta_t$ is replaced by a random displacement with nonzero mean $\alpha,$ we have a "random walk with drift."  An alternative way to express this is to write $\delta_t = \alpha + \epsilon_t$ where now the random term $\epsilon_t$ has zero mean, whence model (2)
$$y_t = \alpha + y_{t-1} + \epsilon_t\tag{RW}$$
is a random walk with drift.  (It is also clear that it is a special case of an AR$(1)$ model with $\phi_1=1,$ but since the problem specifies that $|\phi_1| \lt 1,$ it rules out this characterization.) 

Although we do not have data, we can simulate such models by choosing appropriate starting values at time $t=0$ and generating subsequent values as prescribed by each one.  I thought it might be helpful to see such simulations. Here is an example using the R code below.  All four plots use exactly the same white noise series. You may modify the parameters in that code to see other examples of these models.
Because it is useful to see what an unusual value, or "impulse," does to these models, I have modified one of the noise values to make it unusually large.  The time of modification is marked by a vertical red line in each plot.

#
# Specify data length and parameter values.
#
n <- 192      # Series length, including initial value
alpha <- -0.1
rho <- 0.9
mu <- -0.9
y.0 <- 0      # Initial value of y
dy.0 <- 0     # Initial value of the first difference of y (ARIMA only)
#
# Generate the random terms.
#
set.seed(26) # 17 gives qualitatively unusual plots
eps <- rnorm(n-1)
eps[floor(n/2)] <- 10 # Show what an "impulse" does
#
# Generate time series.
#
y.AR <- Reduce(function(y, e) alpha + rho*y + e, eps, init=y.0, accumulate=TRUE)

y.RW <- Reduce(function(y, e) alpha + y + e, eps, init=y.0, accumulate=TRUE)

e.MA <- eps + mu*c(0, eps[-(n-1)])
y.MA <- alpha + c(y.0, e.MA)

dy <- Reduce(function(y, e) alpha + rho*y + e, e.MA[-(n-1)], init=dy.0, accumulate=TRUE)
y.ARIMA <- cumsum(c(y.0, dy))
#
# Plot them.
#
models <- list(Autoregressive=y.AR,
               `Random walk`=y.RW,
               `Moving average`=y.MA,
               ARIMA=y.ARIMA)
par(mfrow=c(2,2))
for (s in names(models)) {
  y <- ts(models[[s]])
  plot(y, main=s)
  abline(v=floor(n/2), col="#e0000080", lwd=2)
}
par(mfrow=c(1,1))

A: Try all 4 and select the one with the smallest error sum of squares given that all coefficients are statistically significant.
