yes the above is indeed correct. The reason an object will float is if its total mass is less than the mass of an equivalent volume of whatever its floating in. The fish only float because the water is providing a supportive force that is greater than the gravitational force, ie the mass of the fish is less than the mass of the water they displaced, as previously discussed.
This does not exclude their mass from the total though, as they are part of it. If you have half a bucket of water, and add 1 kg worth of rocks, the total weight goes up by one kg. So why should the same not apply simply because the fish are floating? If the bucket was full of a lighter fluid such as oil then the fish would probably sink (poor fish!).
If you have a bucket of mercury, then iron bolts will float. Wether stuff floats or sinks is just down to the mass of the object in comparison to the fluid its sitting in. Has no effect on total mass. That simple!