I wanted to argue with you for posting this, but you're right, this is a smart way of handling API calls.
It's not really addressing the whole issue of depending on third party APIs if your API is calling on them, but you are technically correct, which I've heard is the best kind of correct.
Definitely. Your website and app should only call your APIs. Your API can then call the third party API from the server. It's a lot easier to change something on the server than having to update the app.