What is the difference between these two, and are they both actually the population linear regression model? I am trying to understand the difference between these two stated population linear regression models. My professor's notes state that, if we have predictors $X_1, \dots, X_p$, then the population linear regression model is $Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \dots \beta_p X_p + \epsilon$. But I have also seen the population linear regression model stated as $Y_i = \beta_0 + \beta_1 X_i + \epsilon_i$. What is the difference between these two, and are they both actually the population linear regression model?

EDIT
If you Google "linear regression population model," you'll find these images:




The linear regression model is
$$y_i = \beta_0 + \beta_1 x_{1i} + \cdots + \beta_p x_{pi} + \epsilon_i$$
for $i=1,\ldots,n$ and $\epsilon_i \sim N(0,\sigma^2)$ and independent
Here $y_i$ is the outcome (or dependent variable) for unit $i$ (e.g. person $i$). Sometimes this indexed is dropped and simply stated as
$$y = \beta_0 + \beta_1 x_{1} + \cdots + \beta_p x_{p} + \epsilon$$
"simple" linear regression is linear regression with just one covariate in which case you have
$$
y_i = \beta_0 + \beta_1 x_i + \epsilon_i
$$
Note we can drop one of the indexes on the covariate $x$ to indicate which of our covariates it is, since there is only 1.
The "population" linear regression isn't a well defined term. Probably most people would assume this means the population mean in the above linear regression which would simply be
$$\mu = \beta_0 + \beta_1 x_{1} + \cdots + \beta_p x_{p}$$
Additionally, linear regression can be expressed more compactly in matrix notation as
$$
\mathbf{y} = \mathbf{X}\mathbf{\beta} + \mathbf{\epsilon}$$
Where $\mathbf{y}$ is the $n\times 1$ vector $\mathbf{y}=(y_1,\ldots,y_n)$, $\mathbf{X}$ is the $n\times p$ matrix
$$\begin{pmatrix} x_{11} & \cdots & x_{1p}\\
& \ddots & \\
x_{n1} & \cdots & x_{np} \end{pmatrix}$$ and $\epsilon$ is the $n\times 1$ vector $\epsilon = (\epsilon_1,\ldots,\epsilon_n)$
