Is it true that all contagious diseases are not infectious but all infectious diseases are contagious?

The other way around: All infectious diseases are not contagious. By definition, "contagious" means being communicable by contact. For instance, rabies is not transmitted by usual contact with infected patient.

No: You have it completely backwards. Almost if not all contagious diseases are caused by infectious microorganisms. Many infectious diseases such as e. Coli urinary tract infections are not contagious

Contagious disease: By definition - contagious means you're liable to 'catch' something from someone else. I think of 'the flu', chickenpox, 'meningitis' as good examples of that. (So always good to practice hygiene!). Whereas infectious diseases encompasses all that PLUS other diseases caused by 'germs' that may not necessarily be contagious: for example those deadly amoeba that kill kids every summer, Lyme dz etc.