Your first one is a fabric hammock. The second is a rope hammock with spreader bars.
I don't know that there are any more specialized terms for these. I wouldn't use "sagging hammock" although if you must, I suppose you could use "slack hammock." Still, all hammocks form the familiar curve shape when someone is in them, so I don't think I'd talk too much about their shape when filled.
Normally, hammocks are described by either style, e.g. Mayan, or material, e.g. fabric, rope, string.