Phase angle and transfer function for low pass RL circuit
The natural frequency is defined as:
$$ \omega_c = \frac{R}{L} = 2 \pi f_c$$
\$ \omega \$ in the transfer function and \$ \omega_c \$ are both in rad/s. So the same frequency means \$ \omega = \omega_c \$. There is no \$ 2 \pi \$ in there.

Secondly, when you calculate the amplitude, you need to take the square root.

So overall,
$$ \vert{T}\vert = \sqrt{\frac{1}{1 + {(\frac{\omega}{\omega_c})}^2}} $$
when \$ \omega = \omega_c \$,
$$ = \sqrt{ \frac{1}{2} } = 0.707 $$