Source transformation with impedance being inductor or capacitor
In this case the error was mine. As @The Photon noted, I'm mixing phasor notation (frequency domain) with time domain notation.



In time domain, we have (as pointed in the question):

$$i_s(t)=cos(\omega t)$$

$$e_s(t)=L\dfrac{d(i_s(t))}{dt}=-\omega L \cdot sin (\omega t)$$



Going from the time domain to the frequency domain (phasor notation):
$$i_s(t)=cos(\omega t) \iff I_s=1 \space \angle \space 0 $$ 



In frequency domain, we have:
$$I_s=1 \space \angle \space 0 $$
$$E_s=Z_L \cdot I_s=j \omega L \cdot 1 \space \angle \space 0$$

Remembering that: 
$$ j\omega L = \sqrt{0^2 +(\omega L)^2} \angle \space atan2 \space (\omega L,0)
 = \omega L \space \angle \space \frac{\pi}{2}$$
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(Note: here I am converting from the rectangular phasor form to the polar phasor form)

Thus, we can write:
$$E_s=Z_L \cdot I_s=\left (\omega L \space \angle \space \frac{\pi}{2} \right)  \cdot \left( 1 \space \angle \space 0 \right)= \omega L \space \angle \space \frac{\pi}{2}$$



Going from frequency domain (phasor notation) to time domain:
$$ E_s = \omega L \space \angle \space \frac{\pi}{2} \iff e_s(t)= \omega L \cdot cos \left( \omega t + \frac{\pi}{2} \right) =-\omega L \cdot sin(\omega t) $$



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Therefore, both methods produce the same result.



Note - Remembering that from frequency to time domain we have:
$$ j \omega \iff \frac{d}{dt} $$ 

We can see that:
$$E_s=Z_L \cdot I_s = j \omega L \cdot I_s = L \cdot (j \omega I_s) \iff L\frac{d(i_s(t))}{dt}=e_s (t)$$