The Chemical formula for Diamond is "C". "C" is also the symbol for the chemical element Carbon. Diamond is an allotrope (pure form of same element, that differ in structure) of Carbon.
For example, Graphite is an allotrope of Carbon too. The only difference is the arrangement of atoms, that leads to change in their physical properties.
Diamond - One carbon atom is attached to four other carbon atoms
Graphite - One carbon atom is attached to three other carbon atoms
So you are right! Diamond is pure carbon (chemical symbol "C"), that has gone through intense heat and pressure. This transformed material is diamond and there is no other scientific name.
I hope this helps.