Confused on the interpretation of regression coefficients Let's suppose we have the following regression model:
$$Y_i=\beta_0+\beta_1D_i+\beta_2D_iX_i+\epsilon_i$$ where $Y_i$ represents the test score of the i-th student, $D_i$ is a dummy variable that takes the value 1 if the student's family is an immigrant and 0 otherwise, and $X_i$ is also a dummy variable that takes 1 if the student's father has an above average income.
Now if I were to describe what the coefficients $B_0,B_1,B_2$ mean how would I go about doing that?
I believe $B_0$ is simply the slope. $B_1$ is the change in the mean test score ($Y_i$) when comparing a family who is an immigrant with a non-immigrant. $B_2$ is a bit more difficult. The answers I have here are that $B_2$ is just the effect of father's income on immigrant students' average test score.
I'm a bit confused as to how to correctly interpret these coefficients (especially when there is interaction) and help would be greatly appreciated!  
$\bf{EDIT}$:
Here is an original picture of the question since it appears as though my professor's model is a bit out of the ordinary.

 A: You have 3 different types of students described by the regression equation. 


*

*non-immigrants

*immigrants without fathers earning above average income

*immigrants with fathers earning above average income


Consider the value of $\beta_0+\beta_1D_i+\beta_2D_iX_i$ in each of the above cases as it will help you interpret the coefficients.
A: You can lay out different scenarios in a table like this:

The 1 and 0 indicates if the regression estimates will be included (1) or excluded (0)
Since there isn't an indicator just for father's income, both non-immigrants with either rich or poor father were grouped together, represented just by the intercept, $\beta_0$.
Now, if we recast this into a 2x2 table, it'd be easier to look at:

Pay attention to which two groups that are only differ by $\beta_2$, and those are the groups whose mean difference $\beta_2$ is intended to capture. Hints: focus on the last column.
A: If your model were as follows:
$$
y_i = B_0 + B_1*\text{Immigrant}_i +B_2*\text{UpperIncome}_i +B_3*\text{Immigrant}_i*\text{UpperIncome}_i + e_i, 
$$
where $\text{Immigrant}_i$ and $\text{UpperIncome}_i$ were 0,1 dummy variables, then $B_1$ would be the effect of immigrant status when $\text{UpperIncome}$ status = 0,  and the effect of immigrant status is $B_1 + B_3$ when $\text{UpperIncome}$ status = 1. Similarly, $B_2$ would be the effect of $\text{UpperIncome}$ status = 1 when Immigrant status = 0, and the effect of UpperIncome status is $B2 + B3$ when Immigrant status = 1. If there is no interaction ($B_3 = 0$), then the effect of Immigrant status is independent of UpperIncome status and vice versa.
