What's the difference between carlist and legitimist?

Carlist

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Definition:

• (n.) A partisan of Charles X. of France, or of Don Carlos of Spain.

Example Sentences:

Legitimist

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Definition:

• (n.) One who supports legitimate authority; esp., one who believes in hereditary monarchy, as a divine right.
• (n.) Specifically, a supporter of the claims of the elder branch of the Bourbon dynasty to the crown of France.

Example Sentences:

• (1) The Front National is a family party, and very legitimist.