Hi Sparx,
If a SMAW electrode has a core wire that is substantially different than the weld deposit, such as mild steel, the electrode will be classified as either a EXXX-25 or EXXX-26. The electrodes classified as EXXX-15, 16, 0r 17 do have stainless steel core wires, but a lot of manufacturers only stock a few grades of wire and make the rest of the alloy up in the coating as you suggest. 304L is probably the most common core wire, and as it is drawn down in diameter and cold worked, a lot of it transforms to martensite, which is magnetic. Some people mis-interpret this as being not a stainless steel core wire. The -25 and -26 electrodes do have the advantage of being able to carry higher currents though, and thus higher deposition rates, but are only recommended to be used in the flat or horizontal positions.