On scene command us generally approached as a unified command....meaning Fire and LE working together, usually by hanging out at whichever person has the better positioned or more advanced/useful command vehicle. EMS is generally established as a branch of that command.
With the exception of gigantic disasters, the EOC skills are usually most employed by dispatchers, because dispatch (especially progressive county-centered services) are essentially an EOC. On-scene command is established, but again.....with the exception of gigantic disasters or pre-planned huge events (Fairs, sporting events, marathons, etc) it usually consists of a chief's truck and a pull out tent and whiteboard/scoreboard, if they even get that far. Calling such an EOC is not appropriate.
Note: My experience is colored by working as an EMS chief as my past job in the northeast where I worked a few MCIs. My current agency has ops/EOC centers with varying levels of staffing, and I've never worked in one outside of hands on training exercises. We don't really follow ICS/NIMS either due to a completely different mission.
ETA: Most professional EOC positions are usually called "watch standers", or possibly "desk officer". Most full-time professional EOC positions are located in the federal government, precious few in state governments and very large cities. #JDD might be able to help with delving deeper into EOC/ops centers that he has worked in/with.