"Dirt Track racing" is an american term but even that has changed its now "Flat Track racing".
However the term Grass track racing poses a few questions , for many of the early trotting tracks (Australia & New Zealand) had grass surfaces including Maitland. Both the Blackpool and Greenford Trotting tracks had grass surfaces ? so was it Grass track racing?. Grass racing on oval circuits (in England) was the result of the speedway boom of 1928. Many Motorcycle clubs keen to cash in on the speedway boom organised oval grass tracks or Grass speedways as were sometimes known. Prior to 1928 Grass track racing meant something entirely different, no broadsiding, tracks with left and right hand bends of different sizes and shapes, same with the straights., hillside venues were known as Mountain Grass tracks, when the top surface wore out they sometimes laid gravel, now we have "path racing", these grass tracks were also known as Miniature TT, they were the precursor to circuit racing, Brands Hatch being an example.
There are examples of beach racing (Oval sand tracks) in England prior to 1928, very very long straights with very tight bends and there are photos of riders broadsiding the corners.