Java is compiled into a specific form of byte code
Yes, it's for java, not for any other language. At least I haven't heard you could compile some other language to java byte code and run with jvm (well, except Scala I guess?). So the java byte code IS java, just in a sort of half-ass-compiled way, slower than pure machine code.
Java byte code is Java only in the loosest possible definition of that word. You wouldn't look at an executable compiled from a C++ program for a Windows machine and say, "Oh, that was written in Windows. I don't do windows."
It wasn't written in Windows, it was written in C++, it just happens have been compiled against the windows environment. JVM is the same. It's an environment you can compile code against. I can't think of any reason you couldn't compile any language against the JVM environment (though I can think of several reasons why you shouldn't compile done languages against JVM).
JVM allows you to write once, play anywhere. That's why it's a "half-ass-compiled" sort of thing. The local JVM interprets the specifics of X vs. DirectX vs. iDisplay (no clue what apple calls their display suite) or various file system things or other interoperability concerns. It is slower, a fact highly unlikely to matter unless you've implemented particle physics in your game (in which case, don't do that. The world doesn't need 2 Dwarf Fortresses) because it's pretty damn fast. For reference, Minecraft is written in Java. It runs just fine speed wise.