Question about Arctic Ice Melting

Hopefully we some amateur scientist on here who can tell me if I am correct about the following.

If I chuck a great big block of ice into a bath tub which is a quarter filled with water the ice will displace the water and the level in the bathtub will rise. It normally understood that 90% of an iceberg is hidden below the surface of the seas and I assume the same would apply to my block of ice. With me so far? OK. The ice melts. What will the effect be on the water level in the bathtub? Common sense tells me that there would be very little if any difference in level as ice is less dense than water, which is why it floats.

So to the Arctic. We are currently getting ourselves into an environmental lather over melting sea ice in the arctic, but surely as this ice already floats on the seas the effect on seas levels would be very slight even if it all melted? Have I got this right and if I have why is it never mentioned by the environmental scientists?

This alarmist piece by the BBC would seem therefore to out of all proportion. The real danger, so far as I can see, is land based ice melting and running off into the sea. This would raise sea levels definitely.

For example the majority of ice in Antarctica is over land. Problem is that most projections relating to Global Warming predict an increase in precipitation over Antarctica which will serve to thicken the ice and that thickened ice will also reflect back more of the sun's heat causing a cooling effect.

Complicated this climate stuff isn't it?