Moving magnet and/ or conductor , should they be distinguishable? If the two cases(moving the magnet, or moving the loop) would lead to the same physical consequence(same magnitude of induced emf & current), is it important to distinguish the cause when analyzing similar problems? 
Whether it's from the magnetic force(Lorentz force) or the electric force(from the induced E-field) why should it matter if the results is the same?
I'm curious if there is something important here that I neglected.
Besides being one of the inspirations to SR. 
 A: Provided that the relative velocity of loop and magnet is much less than the speed of light, the emf in the loop in the loop frame (the frame of reference in which the loop is stationary) is the same as the emf in the loop in the magnet's frame. In the first case the emf is due to an induced E field; in the second it's due to the magnetic Lorentz force, as your question implies. As you also imply, Einstein was worried (or stimulated) by the need to employ different methods in the two cases to achieve the same answer. 
When the relative velocity isn't negligible compared to the speed of light, the emf in the loop is greater in the loop's frame than in the magnet's frame by a non-negligible factor of $\gamma$. This is because the time for the magnetic flux through the loop to change by a given amount is a proper time in the loop's frame (events occur in the same place) but an improper (dilated) time in the magnet's frame.
I've written about this in detail in an earlier answer to the question "How to work out the induced electric field of a magnet moving through a solenoid?".
You ask if it's important to distinguish the cause of the emf (electric field or magnetic Lorentz force). I'd say not if the relative velocity is << c. You can work in either the loop's frame or the magnet's frame, as you choose. If the relative velocity isn't negligible compared with c, then you need to choose whether you want the emf in the magnet's frame or the loop's frame.
