Hello @leotohill,
Thank you for joining Auth0 Community and appreciating my article.
Let me try to solve your remaining doubts 
The first diagram is not telling to use ID tokens to call an API when the architecture of your application is a traditional old-fashioned web application.
Sorry if this is what it conveyed to you. Maybe I need to improve its readability.
The two diagrams refer to two different scenarios.
The first one is about authentication; the second one is about authorization.
In the first case, you need an ID token; in the second case, you need an access token.
I hope the reason why you need a certain type of token for each scenario is clear from the article.
Does this help to clarify?