Q. What is the Cosmati method?

Here is a description of Cosmati mosaic by Jane Fawcett in Historic Floors from 1998:

Known as opus Alexandrinum, meaning ornamental paving of coloured marbles arranged in geometrical patterns, Cosmati pavements were composed of roundels, arranged in a guilloche or quincunx design. The guilloche consisted of cirles of marble and mosaic, linked by curved bands of white marble to form a figure S. The quincunx was a figure 5, consisting of a central roundel surrounded by four smaller circles. Within this framework connected by bands and loops of white marble, a complex design of mosaic rectangles, diamonds, circles and squaires ws developed. The whole composition was arranged longitudinally, in order to lead the eye from the west end towards the high altar, for which it provided a setting.

She goes on to talk about the Roman artisans who created these designs:

The Roman Marble workers, or Cosmati, became active in Rome and in many other regions of Italy from the early twelfth century until the decline of the papacy in 1300. With the exile of the Pope to Avignon much of their work ceased, although a few pavements continued to be made during the Renaissance. Some of the Renaissance floors in the Vatican, including the Sistine chapel, are based on Cosmati designs. The craftsmen became known as the Marmorani Romani and were confined to a few families, of which Paulus and his family were the first. Others were Ranucius, Laurentius and Odoricus, who worked at Westminster Abbey. Each family developed its own distinictive style...The pavement of S. Maria in Cosmedin, which resembles the Great Pavement at Westminster Abbey, was designed by Paulus, founder of the earliest Cosmati family. 

She adds the the Great Pavement of Westminster Abbey was designed by Odoricus, a Cosmati craftsman, and completed in 1268. (pp13-15). It seems as if the materials used were mostly semi-precious stones and marble, but in the Great Pavement, according to Fawcett, Purbeck was substituted for marble and glass tesserae for the semi-precious stones. It sounds as if, from this source and a few others with less description, Cosmati or Cosmatesque (for those mosaics inspired by original Cosmati mosaics) refers more to design than technique.

The technique you mention of angling the tesserae was mentioned by several books as a Byzantine practice. According to Elaine Goodwin in the Encyclopedia of Mosaic, "The very early Byzantine mosaic masters of the 6th to 8th centuries AD understood light. They knew that by angling their materials of gold and smalti towards or away from the available light source, they could create surfaces on which light could play to create an ambience...." (p 151). I could not find a name for this technique.

I am attaching a few bibliographies which may be of interest to you on Mosaics.