An Islamist is a political Muslim who seeks to establish Sharia, Islamic Law, as the law of the state, superseding secular law. They are the seekers of an Islamic Republic. Iran is such a state with clerics wielding supreme and ultimate power over the government.
A Jihadist or Jihadi is a militant Islamist.
Radical Islam is any sect or faction of Islam that employs or supplies Jihadis outside of government function.
It is a violation of State Department rules for the US to acknowledge or recognize political entities that do not represent the state, as this represents a subversion of the state government. For example, during the Apartheid era, the State Department could not acknowledge the African National Congress, a party with no seats in South Africa's parliament. So it is reasonable that on a state by state basis we identify all parties involved. This is old news and it is how the US has identified non-state actors like the Taliban.
Where all this gets dicey of course is that it is often general practice for Jihadi organizations to claim support from the 'Street'.