That's not a definition, just a vague statement - sounds more like a definition of mass anyway.
It has to be vague to keep the similarity, but density and mass are related regardless. Mass depends on density and volume: how large an object is, and how much is packed into what's already there.
I get this, but why do things stack in a particular direction? Why aren't we walking on the dome looking down on earth?
See the buoyancy analogy. The dome naturally forms when air gets so far from the Earth's source of heat that it freezes, and forms a full arc. The only surface that air can press against is the Earth, and equally we exist on the Earth's surface, and do so by pushing up. The buoyancy analogy is a good one: we exert a force up, which would be down into the pool in the analogy, and while we can move left and right easily inside the pool without being pushed back, we do get pushed up because that's the direction we displaced the water from.
Heat? This is new to me....can you explain?
Essentially molecules under Scepti's models are balls which can form jawbreaker-type combinations when they fit inside one another, but to do that they require energy to compress them (sponge balls in a fist was Scepti's analogy). This source of energy is heat. So, near the part of the Earth that we know, and the Sun, there's enough energy for this to happen. Going up, there's less heat, and less energy compressing molecules, so they all spread out, meaning the lighter elements are up there.
Riddle me this: I went for a hike the other day - 10 miles from inland to a cliff. I'm standing on a cliff and the air pressure is equal all around me. I then stepped off the cliff (the air pressure is still equal) and plunged into the sea. Why did this happen? Why didn't I "fall" towards the dome, or shoot off sideways?
Because you push air up: much like how if you go into a pool buoyancy only acts in one direction, opposite your direction of entry, regardless of how much you move left and right etc. Buoyancy does rely on gravity as we're familiar with it, but ultimately gravity does just act like a force, and can be replaced by any force, such as a basic your-mass-accelerating-up.