How do I update to the latest version of safari? On App Store there's no update available? I have an older Macbook pro.

NB the footnotes in your linked article:

Update: A later version of the currently installed software. Learn about macOS Big Sur updates, for example.

Upgrade: A major new version, often with new system requirements. Each macOS upgrade has a unique name, such as macOS Big Sur. Learn more about installing the latest macOS upgrade. Some old versions of macOS are also available.

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If you need more help, Please post the model-year of your MacBook Pro and its screen size, as well as the version of MacOS you are currently running.

Depending how old your computer is and OS, you may not have an update for Safari available. There is a point that the computer OS can't be updated because it doesn't have the capability to run the newer OS, which may also be required to run the newer version of Safari. if this is the case, you would need to get a newer model of the MacBook Pro. You didn't provide any specifics on age of computer, OS being used or the version of Safari you have to give a specific answer to your question.