The record of the National Fruit Collection, at Brogdale in Kent, describes it1:
Raised in about 1832 by Liegel at Braunau, Austria. It was named after Schmidtberger, a pomologist at St Florian monastery. Fruits have firm, greenish white flesh with an acid flavour.
Where Schmidtbergers Rote Reinette came from.
Sources
Schmidtbergers Rote Reinette (accession 1951 - 050), National Fruit Collection, Brogdale, Kent (Defra).
Supports: the description quoted; the use; the fruit's size, shape and colours and the flesh, from the collection's characterisation; the picking time at Brogdale (early october); the other names.