Refactor error handling examples for the ts-graphviz/adapter usage

This issue is a follow-up for the concern mentioned here: https://github.com/ts-graphviz/ts-graphviz.github.io/pull/36#discussion_r1898844203.
User: kamiazya has requested to create an issue to track the necessary corrections throughout the document.

Pull Request URL: https://github.com/ts-graphviz/ts-graphviz.github.io/pull/36