Firstly, I'm not a native speaker but from personal experience the word "import" is most often used in the meaning "A commodity, article, or service brought in from abroad for sale" and less often used in the meaning "The implicit meaning or significance of something".
As synonyms "import" and "importance" aren't quite interchangeable but share common meanings. The word "import" carries more than is given by the definition you refer to. From practice, the word "import" implies several definitions (especially 'of import'):
- intended to bring significant value
- has important effects or influence
- a meaning that is not expressly stated but can be inferred
Thus knowing this we can understand the following examples as:
- He's the guy of import in the company. - He's the one who brings significant value or (and) exerts his influence in the company
- Nothing of import, My Lord. - Nothing significant
- A quick import of his speech was ready in ten minutes. - An implication of his speech: the meaning that is not expressly stated but can be inferred
- The raise of import of the company in the area comes from its employees. - The company gets its influence raise in the area from the good work of its employees.