Including level 2 covariates at level 1 in multilevel modelling

I'm familiar with multilevel modelling. However, I'm wondering what would happen if you would include level 2 covariates at level 1. If you have, for example, students at level 1 (age, gender,...) and schools (living area, province,...) at level 2 and your dataset looks like this

studentID age gender schoolID living_area province  Y
1         12    1       1        1           1     2.4
2         8     0       1        1           1     3.1
3         10    0       2        0           0     5.2
4         12    0       2        0           0     3.9
5         10    1       3        1           0     4.1
6         9     1       3        1           0     4.8


Why can't I write my model as follows:

$$Y_{ij} = \alpha_j + \beta_1*age_{ij} + \beta_2*gender_{ij} + \beta_3*living\_area_{ij} + \beta_4*province_{ij}$$ $$\alpha_j \sim N(0, \sigma^2_b)$$

And why is it better to write my model as follows:

$$Y_{ij} = \beta_1*age_{ij} + \beta_2*gender_{ij} + \alpha_j$$

$$\alpha_j = \alpha_0 + \alpha_1*living\_area_{j} + \alpha_2*province_{j}$$

What are the advantages of modelling it that way?

You could go with a mixed formulation, which substitutes the level 2 equation into the level 1 equation:

$$Y_{ij} = \alpha_0 + \beta_1*age_{ij} + \beta_2*gender_{ij} + \alpha_1*living\_area_{j} + \alpha_2*province_{j}$$

$$\alpha_j \sim N(0, \sigma^2_b)$$

This correctly signifies that living area and province are measured at the upper level of your data hierarchy (the $$_j$$ subscript) and shows the common intercept ($$\alpha_0$$).

Is your question mainly about the model equation or are you also wondering what would happen if you ran a single level OLS regression that included the level 2 predictors in the model?