In case the organisation is basically a tenant in multi-tenant application I would leave it out of the API and resolve it in another way, but if it is not I would leave it in. Does that make sense?
In some cases there might be a next stage of complexity where you have users with access to multiple tenants who are browsing them in different tabs. Then putting the organization as the top level in _every_ URL helps signal intent; otherwise you have to take other precautions that someone isn't on a tab with Organization A, clicks "Post new article" button (-> POST /articles) and happens to have their session logged…
I'd really recommend putting the organization name in the content of the page in that example. Users don't really look at URLs, browsers de-prioritize them and often only display part of the URL.
Also, depending on front-end implementation the REST API URL may not be displayed in the URL bar. A single page app may use example.com/#create-article for posting an article. A mobile app wouldn't even have a URL bar.