Adopt new MarkDown alert extension in documentation

[ericcornelissen]

Summary

GitHub recently released a blog post announcing the adoption of a new MarkDown extension for displaying alerts (documentation). This issue exists to track the updating of the documentation to adopt this new extension.

To get an idea of the scope of this issue, most instances of existing "alerts" in the documentation can be found using this GitHub search query. (If you work on this locally, searching the entire codebase for the string > ** should give the same result.)