I think that it is fairly conclusive that the British educated and noble class did consider themselves Roman. They took Roman names, set up monuments in latin and in the period after 410 still considered themselves Roman, and were indeed that last free Western Romans. They supported several emperors of their own after all.
The landscape around me where I am writing is dotted with the remains of romanised estates deriving from pre-Roman Iron-Age estates where nobles and kings settled down to a Roman way of life. Just a few hundred yards from where I work is one of the largest villa complexes in Britain, certainly the estate of a king of some sort, and indeed the settlement later became the regional centre for West Kent, even though it is only a small village now.
After the legions had left Romanitas continued. St Illtud's monastery provided a full classical education, St Columbanus writes in the best Latin, even St Patrick writes Latin despite his education having been cut short.
There are pilgrim flasks deposited in 5th-6th century sites from the great centre of Aba Mena in the Egyptian desert. The British got around and were not parochial by any means.
As Ken Dark, Charles Thomas and many others show, Romanitas didn't end in 410, and its continuance shows that despite economic collapse after 410 the British did their best to keep up appearances. Think of Ambrosius Aurelianus, think of all the crosses and monuments with Latin inscriptions and with 5th century figures maintaining the appearance of participating in a civil society. Even Wroxeter with many buildings being constructed after the legions had left, certainly not in the same style as in the height of the empire, but still consciously trying to be Roman and civilised.
The West, like the East, exhibited a variety of means of being Roman - which surely means civilised as much as anything else.