Is best known for taking older forms and styles and redefining them in modern ways Antonín Dvorák was inspired by his Hungarian Dance No 1 What composer is this?

The "mystery composer" you are asking about is Johannes Brahms, although it is stretching a point to say that Brahms took older forms and styles and redefined them in a "modern" way.

Brahms certainly had his own voice, but no musicologist I have ever heard of would describe Brahms as an "innovator". He used existing forms in fresh ways, yes - but that's originality, not innovation - they are two different things. Brahms was a fine composer, but he was no trailblazer. His First symphony - and it is magnificent - was called "Beethoven's Tenth" for a reason.