Are there lava flows inside a volcano?
This is largely an issue of trying to show three dimensional structures in a two dimensional medium. It is also an issue of trying to differentiate between different materials and volcanic behaviours.

Volcanoes can be composed of three types of material: Country Rock is the material they're erupting up through it's usually sedimentary near the surface and gets buried under volcanic rocks relatively quickly, so much so that few diagrams include it. The other two rock types are volcanic in origin these are Cinders and Lava. Cinder is material thrown out of the volcano, this is usually Scoria but can include other Tephra as well, importantly it lands as a solid. Lava is the liquid rock that flows continuously from a volcanic vent. These two volcanic rock types are usually mutually exclusive, eruptions with lava have few cinders and vice versa.

There are three basic "plan-forms" of volcano Basalt Shields, Cinder Cones, and Composite Cones.


Basalt Shields form slowly from layer upon layer of lava flows that solidify one atop another over long periods of time they are correspondingly flat. 
At the other end of the eruptive spectrum are steep Cinder Cones formed from more violent eruptions that throw a lot of material out of the volcano, that rock solidifies as it flies through the air and lands around the vent. The closer you are to the vent the more material there is landing and the bigger it tends to be so a cone builds up, the angle of it's sides being governed by the average size of rock landing at that point. 
Composite Cones are exactly what the name says; cones built up from a combination of Cinder and Lava from separate eruptions, some gentle and others violent. 


The diagrams you've presented are an attempt to differentiate between these three plan-forms based on the nature of the volcanic material involved, using a deceptively simple palette. In active volcanoes there usually are lava distribution systems that move laterally through the visible cone, these feed lateral vents or lava tubes moving material far from the central vent.