a deductive argument form and rule of inference

C \to can be shown that with only these three axiom schemata and '' modus ponens In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for 'method of putting by placing') or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference.

an inference rule that allows us to deduce new propositions from old ones

The difference is that this tautology is a single proposition, whereas modus ponens is an inference rule that allows us to deduce new propositions from old ones.

a form of valid inference

Modus Ponens (Latin: mode that affirms; often abbreviated as MP) is a form of valid inference.

a famous rule of logic

Modus ponens is a famous rule of logic used in natural deduction.