I grew up in the southeastern U.S., using a stove with the old fashioned electric coil burners. We always called them "eyes."
As an adult living elsewhere, I continued to call them eyes as long as I had that type of stove. Then I had gas ones and called them burners. Then I had the flat electric ones (which I HATE, btw, they don't get hot enough and I can't keep them clean) and I never know what to call them. It's usually "the eyes, burners, whatever you call those things."
I just noticed in my first sentence that apparently I've started calling the electric coil ones burners as well.
Total side note: That's one of the hazards of living in a lot of different places and being exposed to lots of different dialects and languages ... sometimes one word comes out naturally in one environment but a different word in another environment, or the word that comes out most naturally isn't even your native language (if it's something that there is an English word for but I didn't experience much until I was in a non-English speaking country and I just used the local word for it--for me, a prime example is a traffic circle or roundabout; sometimes I use those words, but the most natural one is the Arabic "midan") or two or more words try to come out at the same time and I just stop, confused, while I figure out which one to say.