Your correct, the documentation is probably a little out of date.
For oEmbed, you need a Facebook application. In our documentation, we also mention you need the Instagram Basic API access, but that is incorrect now. We'll get that updated.
As for oEmbed, when you have the Facebook developer app and secret setup, you should be able to use the oEmbed widget.
However, please note that the oEmbed widget allows you to share 1 post at a time. When you configure it, you'll set a URL or ID to a specific Instagram item to show on your site. It cannot be used to show a feed of all of your posts. Instagram does not have an API to allow pulling your feed and displaying it. We do support feeds for Facebook Pages, Twitter Streams and LinkedIn Company Feeds.
I hope that helps explain, but if you have any other questions, just let me know.
Alex