Yes, the water level needs to be no lower than the red line shown on the attachment below. At least when starting, or when running in a bucket. During normal operation when the motor is running through the water and the boat is up on a plane, water is getting scooped by the pickup on the underside of the exhaust, downstream from the prop.
You would have damaged the impeller withing 5 seconds of running, but no matter. Given the age of the one that is in there now it needed to be changed anyway. Check to make sure it hasn't disintegrated, and if it has, account for all the rubber bits (you don't want them clogging a water tube).