This is documentation for the API.

What does "Please specify <kbd>format=json</kbd> explicitly." mean here? Where and on what occasions should the user specify this kbd?

If users of the API do not specify format=json in the request, the system won't work (welcome to MediaWiki's APIs).

https://translatewiki.net/wiki/MediaWiki:Apihelp-templatedata-example-1
https://translatewiki.net/wiki/MediaWiki:Apihelp-templatedata-param-lang
Documentations do not match message content for these two. Please specify the context of these two messages.

I have no idea why the "doc-apihelp-example" template on TWN is written that way, but this isn't a task we can fix. I'd suggest talking to other TWNers?