Edit: Now that the question has been modified following my original answer, which is kept below, my modified answer is:
What's the meaning of indexes in the definition of populational regression?
The usual meaning is to index the observation number, so that $Y_1$ and $X_1$ refer to the 1st observation of the variables $Y$ and $X$. In general $Y_i$ and $X_i$ index the $i$th observation.
So the models:
$$Y_i=\beta_0+\beta_1X_i+e_i$$
and
$$ Y=\beta_0+\beta_1X+e $$
are entirely equivalent. The only difference is that in the former we are dealing with individual observations, while in the latter we are dealing with the variables themselves as vectors.
The model:
$$ y_i=\hat{\beta_0}+\hat{\beta}_1x_i+\epsilon_i $$
would usually be used when discussing predictions from either of the models above, though for consistency I would prefer to keep the same upper or lower case notation. $\hat{\beta_0}$ and $\hat{\beta_1}$ are then the estimated ceofficients obtained after fitting either of the above models.
Original answer:
Different people use notation differently. Commonly, the subscripts refer to a particular observation, so
$$Y_i=\beta_0+\beta_1X_i+e_i$$
and
$$ y_i=\hat{\beta_0}+\hat{\beta_1x}_i+\epsilon_i $$
should be equivalent (except that the hats usually represents an estimated value, rather than being used as the description of the model). So $Y_1$ and $X_1$ refer to the 1st observation of the variables $Y$ and $X$. Similarly for $y_i$ and $x_i$: they represent the $i$th observation.
On the other hand,
$$Y=\beta_0+\beta_1X+e$$
could be used when $Y$ is a vector of responses and $X$ is a vector representing a single explanatory variable. It is up to the person writing it out to be clear about what notation they are using and what it means. When dealing with vectors and matrices in regression it is also common to write:
$$Y=X\beta +e$$
where $X$ is a model matrix which will include all explanatory variables including the intercept, and $\beta$ is a vector of coefficients.
