There are some different regions and you need to assign them all names and keep them straight or you'll get them confused.
There is the outside. If you had a solid conductor without a cavity it would be the part that isn't the conductor.
There is the cavity. It's like a hollow region inside the conductor, surrounded by the conductor. It's like the inside of your house if you closed all the doors and windows. It's not the house itself, it's not the walls, it's not the floors or ceiling or roof. It's the region inside with all the air and people. If you replaced the entire outside with a conductor it's the part that now isn't a conductor.
Then the third region is the conductor itself. It's the region that isn't the outside and also isn't the cavity. It's a conducting metal. It has zero electric field.
What about the other regions, do they have zero electric fields. They don't have to, and often they do not.
The field lines come out of the charge in the cavity, go through the region of the cavity and end on the inside surface so they only exist inside the cavity, not in the conductor. Recall those are two different regions.
Next if there is charge on the outside surface the field lines come out of those charges and head out into the outside region. So they exist only on the outside, not in the conductor. Recall that those are also two different regions.
But isn't outside just the outer surface of the conductor? Why make it so complicated? The book just says that all the net charges will stay right on the surface.
It's not overly complicated to have different names for different things so you can tell when people are talking about different things.
There are two regions of not-conductor and one is called the outside and one is called the cavity. The surface between the outside and the conductor is called the outside surface. The surface between the cavity and the conductor is called the inside surface. Both surfaces are on the boundary of the conductor. Neither is "inside" the conductor in the sense of being anywhere other than the boundary between the conductor and the not-conductor. You simply need to learn the terminology to learn what people are saying.
Your post clearly showed you had trouble distinguishing when people were talking about the cavity, when they were talking about the outside, and when they were talking about the conducting shell. There is charge on the inside surface (between the cavity and the conductor) and the field lines in the cavity end on the inside surface. They don't go into the conductor. You claimed there was a contradiction, but it's just you misunderstanding when they are talking about each of these three regions. Only one region (the conducting shell) has zero electric field.