What hooks do I need to hook into to capture ever wp_user creation/change?

Those first 2 actions you linked to from the codex, user_register and profile_update should be exactly what you need.

user_register runs right after a user has successfully registered and gives you access to their $user_id which you can use to retrieve any user data you need, like name, email, etc.

add_action( 'user_register', 'wse_custom_register_action', 10, 1 );
function wse_custom_register_action( $user_id ) {
    // send data to your external db/API
}

profile_update works exactly the same way, and gives you access to both the old user data and the new, updated user data.

I don’t believe you would need edit_user_profile_update in this instance.