Membership Auto-Renew Changes in AuthorzeNet
Changes made to recurring contributions directly through the payment processor portal tend to break the communication back to Civi on subsequent installments (as you have surmised).

For any significant changes to a recurring contribution (amount, credit card type, financial type, etc.), it is best to cancel the existing recurring subscription and create a new one to avoid communication and other issues. If all you need to update are the credit card details (e.g. someone wants to use a different AMEX card from the one they originally created the subscription with), then making those changes from within Civi should communicate with Authorize.net without interrupting communication back from authorize.net on subsequent installments. You would do this by clicking the "More" hyperlink to the right-hand side of the recurring contribution record, select "Change Billing Details" and entering the details. The best way to see if this works is to update the credit card details directly from Civi, then log into the authorize.net portal and check to see that the updated information is listed on the subscription in question.

Hope this helps!
Tamar