output resistance of BJT
It is not a surprise. Notice that in rule 3 you neglecting not only the Early effect (\$r_o\$).
But also a fact that:
\$\frac{1}{g_m}\$ &lt;&lt; \$r_\pi\$
Aditional notice that
$$\frac{1}{g_m} || r_\pi = \frac{1}{ g_m + \frac{1}{r_{\pi}}} = \frac{r_{\pi}}{g_mr_{\pi} +1} $$
Also, did not forget that :  \$ g_mr_{\pi} = \frac{I_C}{V_T}\frac{V_T}{I_B} = \frac{I_C}{I_B} = \beta\$
Therefore the impedance looking into the emitter is:
$$\frac{1}{g_m} || r_\pi = \frac{r_{\pi}}{\beta +1} = r_e $$

$$r_e = \frac{dV_{BE}}{dI_E} = \frac{V_T}{I_E} = \frac{r_\pi}{\beta +1} = \frac{\alpha}{g_m} = \frac{\beta}{g_m (\beta +1)} $$
But most a time we use a simplified approach thus, \$I_E \approx I_C\$ we can simplify our life and assumed that:
$$r_e \approx \frac{1}{g_m}$$