Hi, welcome to JustAnswer!. This is Ed.
This looks like a chicken-or-the-egg or a he said, she said sort of problem; the CE light is on, but we don't know if it's being commanded to be on by the PCM (engine controller) or if it's a logic mistake by the instrument cluster. Ultimately, the CE light is controlled by the PCM with its intentions communicated through the vehicle's bus network (the CCD bus), but the cluster always has the capacity to close the circuit and illuminate the bulb, whether it's being told to or not. An in-depth scan of the vehicle might reveal the cause for your CE light illumination if the scanner can look at inputs-outputs for PCM and cluster. Each system will display an intent for the state of the CE light, which will be either on or off. If the PCM says on, it would fit that the cluster has the light turned on, but if PCM says off, it would be the opposite. If both systems say off... yet it's on... I'd have to go with the cluster as the problem.
Without a scan tool (and it's got to be a factory-caliber unit), you might try unplugging the PCM to see if the CE light goes off. The PCM is conveniently located underhood, next to the positive jump start post and underhood fuse box. It's the aluminum-finned unit with two large 60-way connectors pointing straight up.
With the key off, disconnect both connectors, then roll the key on. If your CE light continues to stay lit, it would tell me that the decision has been made by the cluster to stay on, not the PCM. On the other hand, if the CE light never flickers, the signal is probably coming from the PCM.
I'd just leave the connectors off at the PCM for 5-10 minutes to let it go to sleep at that point and the CE light thing might be a distant memory after reconnection. It happens.