What you want is totally possible, but you'll have to implement it yourself. As the link you found states, there is no built-in on-password-update account trigger... only onCreate and onDelete account triggers. Which means we have to handle it manually. I would handle it the same way you are heading - using a cloud function to send the user an email.
I would build a cloud function named something like notifyUserOfPasswordChange() and call that cloud function from your app immediately after the line of code where you call .updatePassword() or .confirmPasswordReset() (which is the finishing step after .sendPasswordResetEmail()). If I understand the point of your question - this is the real answer here. You will need to call a cloud function manually whenever you execute password update code. There's no automated trigger.
The email can be as simple or customized as you code it. If you're unsure, then start simple and have the cloud function get the target email address from the data parameter - and then use a generic message for the email body. If you feel more adventurous, consider passing the user's UID and using the Admin SDK to look up that user's registered email address & display name and then building a prettier & personalized HTML email.
As far as how to send an email from Firebase cloud functions, there are plenty of examples out there - a good sample is on Firebase's GitHub page.
