Help understanding the integration between Google Voice, Hangouts, and Messenger

Messenger is an SMS application. It can be used to send SMS. Every SMS that are sent from your device are stored somewhere. Other SMS apps can read and write them.

Phone is a phone app. It is used to make phone calls. Every calls made from your device are logged somewhere. Other phone apps can read and write to this list.

Hangouts
is a messaging platform. It uses your
Google+
account to send messages to other Hangouts users via their
Gmail
address. You can also make video calls with it. It is very similar to
Skype.
However, it is also an SMS app! It can send SMS via your phone number and receive messages from other phones.

Hangouts Dialer is a phone app that acts as a plugin to Hangouts. It enables Hangouts to send calls to other phones via your phone number.

The interaction is simple : Your SMS are logged somewhere on your phone and only your default SMS app can access them. This is why some, but not all, messages are displayed in Messenger when you actually sent them from Hangouts.

If you want every messages to be inside the same app, the obvious solution is to keep Hangouts as the default SMS app. However, don't forget to link your contacts infos in
Contacts.