LuaVerge is a sort of silly name given to Verge, when you run in it with Lua scripting enabled. LuaVerge IS Verge, but the name was come up with for it to be really easy to tell which language is being run.
This makes it less effort to talk about features that are specific to the Lua scripting engine in Verge, and probably makes it easier to get help later.
LuaVerge is essentially Verge's core API (from VergeC), but shoved into Lua through a table called "v3". The procedural code that works very well with VC feels a little odd in a Lua world. This is why vx was invented. To package and tidy up the code Verge provides, and make there be some easy-to-use, productivity-enhancing object-orientation.