Accidentally wolfed down a rare burger
There is no way of knowing, from your picture, if there is E. Coli present in your burger, nor is there any way for us to let you know if you will get sick.  However, we can identify the risks and what increases or decreases the risk of infection.  If E. Coli is present, it is typically present on the exterior of a whole muscle.  So, you can decrease your risk of infection by purchasing quality whole muscles from a reputable source, and grinding the beef yourself.  Purchasing pre-ground beef, or pre-made patties, increases your risk, because you don't know the quality of the original cut, or how it was treated during processing.  You can, of course kill E.Coli by cooking, however, you need the relatively high, and generally recommended temperature of 160F (71C), and we now know that some E.Coli survive that.  I don't enjoy my burgers that well done, so I purchase quality whole muscle and grind my own.  Since we don't offer health advice, I can't comment on whether or not you should vomit.