First the good - Thank you for integrating with a two way SMS provider.  Twilio seems like a good solution for that. Some of my users have reported not receiving notifications for inbound SMS messages. The feature is not very useful if there is now way to tell we have received a message.  We found we had to set up a Zapier notification to email our users about incoming SMS messages.  That is not a very good solution.

Also, the notification settings must have a bug in it, because yesterday, I set it to ONLY receive SMS notifications and I am still receiving status changes, etc. Can you look into that?