I believe that the attribution to Luther Burbank is incorrect. Burbank introduced a variegated ornamental corn which he called 'Improved Rainbow' and most likely this is how it became confounded. After searching through lots of his catalogues I can find no rose of that name that he introduced.
From Trumbull & Beebe's Catalogue of Trees and Plants, 1902, page 85
"Improved Rainbow - Is entirely distinct and far superior to Rainbow. The Improved Rainbow is penciled with brightest Gontier color, every petal in every flower and base of petals of a bright amber color, making a very distinct and charming flower."
