Excellent question! There's no definitive way of saying "this is a fossil"; so paleontologists usually pick the arbitary age of 10,000 years - so if remains are more than 10,000 years old, they are called a fossil.
fossil
The Fossils in Pokémon Diamond are the Helix Fossil which is the Omanyte Fossil, the Dome Fossil which is the Kabuto Fossil, the Old Amber which is the Aerodactyl Fossil, the Root Fossil which is the Lileep Fossil, the Claw Fossil which is the Anorith Fossil, the Skull Fossil which is the Cranidos Fossil and the Armor Fossil which is the Shieldon Fossil.
You have 9 types: Armor Fossil, Claw Fossil, Cover Fossil, Dome Fossil, Helix Fossil, Old Amber, Plume Fossil, Root Fossil, Skull Fossil
fossil
Kabuto's fossil is the dome fossil
A turtle fossil
I am pretty sure the dome fossil is kabuto.
either: cast fossil petrified fossil imprint fossil trace fossil
The plume fossil is a fossil that contains the Pokemon archen but you can either get the plume fossil or the cover fossil from this guy in the relic castle.
a helix fossil is a fossil that becomes an omanyte
a full body fossil
The address of the City Of Fossil Museum is: , Fossil, OR 97830