Induction cooker hot on inner circle only
Warping doesn't matter with induction, since you don't need the pan to touch the surface, unlike resistive electric technology. 

I suspect that your unit is working as intended. I have personally never seen home units with a large coil, they typically have a 12 cm coil or less, and since most pans are larger, they only create direct heat in the middle of the pan. You can use larger pans on it because there is thermal conduction happening both within the pan metal and within the food in the pan. My guess is that the outer circle is an indication of what is the maximum suggested size of pan, presumably because the conduction will not be sufficient to produce good results. I have not seen units with an inner circle painted on, but it could be a guideline for the minimum size of pan that the unit accepts. This is very likely if the size of the inner circle is the same as the size of the coil (your burn mark). 

And by the way, your burn mark and your warping both mean that you are using extremely high heat to cook. There are almost no foods which benefit from that. You may consider reducing to medium-high for most cooking - when I had induction, that typically meant using the first or second wattage setting on my unit (cooking for 1 person).