Input resistance confusion
Okay, I think I figured it out.

 

$$i_x = \frac{v_x}{R_E} - \underbrace{(i_b+i_c)}_{i_e}$$

However \$i_b+i_c = (\beta + 1)i_b\$ and \$i_b = -v_x / r_{\pi}\$

Hence

$$\frac{v_x}{i_x} = \frac{1}{\frac{1}{R_E} + \frac{\beta + 1}{r_\pi}}$$