Multilevel modeling: Why aren’t level-1 predictors allowed in level-2 equations?

I’m reading a presentation by Preacher, who gives the following level-1 and level-2 equations

Level 1

$$y_{ij} = \beta_{0j} + \beta_{1j}x_{1ij} + \epsilon_{ij}$$

Level 2

$$\beta_{0j} = \gamma_{00} + \gamma_{01}w_{1j} + u_{0j}$$
$$\beta_{1j} = \gamma_{10} + \gamma_{11}w_{1j} + u_{1j}$$

Preacher notes:

Even though it is not possible to use a level-1 variable as a predictor in a level-3 equation, it is possible for the level-3 intercept variance to be reduced by $$x_{ij}$$. We can explain level-3 variance with level-1 predictors. For example, differences among classes may be partially explained by considering a student-level variable.

I think the reference to level-3 may be a typo, and Preacher actually means level-2, because at this point in his presentation Preacher hasn’t yet introduced the concept of a third level.

Why can’t the level-1 predictor $$x_{ij}$$ be included in either of the Level 2 equations?

A level-1 predictor, $$x_{ij}$$ can't be used in the level-2 equation because there are many different values of $$x_{ij}$$ for a single level-2 unit $$j$$. You can't use, say, individual income as predictor in a model for neighbourhood crime levels, because there are lots of different individual incomes for each neighbourhood.