If a text message is sent via the cellular company controlled SMS or MMS it will register in your cellular text log.
However, as you are finding out, there are many ways to send messages these days. These messages are sent by different programs and companies (instagram, facebook, imessages etc) and are carried over data (cellular data or wifi). These are not controlled or monitored by the cellular company and Verizon has no way of knowing or counting the messages.
That is why, the only safe way to monitor children's use of technology is regular and direct access to his or her phone with the child present to review concerns. If the child knows you are monitoring SMS text messages, they will just message another way.