I think you got it, although the word "restriction" instead of "right" would probably be more accurate when it comes to State and Federal.
Originally, the Federal Constitution only applied to the Federal government. So even though the Bill of Rights said you have freedom of the press, there was nothing in the Federal Constitution preventing New York State from shutting down a newspaper-- although New York had its own Constitution that forbid the State government from doing this.
The 14th Amendment incorporated the Bill of Rights onto the several States. This means that post-14th Amendment New York is prevented by the Federal Constitution from interfering with a free press and that if you believe that they have you can file suit in Federal Court.
As a result if you believe the Federal government has violated your due process rights you sue under the 5th Amendment, and if a State government violates your due process rights you sue under the 14th.
This also creates a situation where Federal due process becomes the minimum standard. A State can grant greater due process rights than the Federal government but never fewer. Same for 4th Amendment searches, and Federal interpretations of the Establishment Clause, etc. (Not trying to say this is good or bad, just the way it currently is.)
OK, fine.