If lava flows back into the ground, would it be called magma?
Lava is per definition the molten material on the surface or a hardened rock thereof. Lava is expelled from a volcano or fissure. There is a rich vocabulary to describe the composition, formation and processes of lava. 

Magma is a melt within the earth. Magma is formed when material reach a temperature for melting or partial melting, e.g. in a magma chamber. Complex high temperature chemistry is used to describe how solid material melts to magma and how it recrystallizes into igneous rocks as a response to temperature, pressure and changed composition. 

Magma can originate from any kind of material. The definitions doesn't refer to the composition or origin of the material, but the manifestation. Magma can be melted or partially melted crust, also old volcanic crust, so if old lava somehow re-enters a magma chamber it will be magma. When magma is erupted, it is lava. 

There is often some confusion, even among professional geologists, about many of the terms we are using. Some terms describes a process and other describes an object. Sometimes we say Upper Jurassic, sometimes Late Jurassic depending on the context. Some terms are historical and some are modern... There are often overlaps. 

As for any language, it's good to use a broad vocabulary and adopt to the jargon, but most important is to communicate the ideas and observations as clear as possible.