In your sample sentences, "the postman" could be a concept, really meaning "mail". It is not meant to mean a particular person.
Has the postman come yet? (Has the mail come yet?)
This morning the postman brought me a letter without a stamp. (This morning, the mail delivered . . .)
You could also say:
Has John, the postman, come yet?
This morning, Mary, the postman, brought me a letter without a stamp.
Here, it means a particular person with a particular job:
John has a job as a postman.
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