It's difficult to pin down national identities in the Middle Ages, since for the most part all we have to go on are texts written by educated individuals who may not be representative of the general sentiment.

With that said, the identity of the people of Wales is likely to have been in a state of flux based on the changing language used to refer to Wales and its people not only by contemporary native writers but by 'literati' all over Europe. The identity from which they were emerging was based on a "British" discourse. They were aware that they were the sole rulers of the former-Roman province before the Anglo-Saxons arrived, and Arthurian legends (in particular, a prophecy that Arthur would return and lead the Britons to reclaim their former land) were popular in the period. They wrote of themselves as being "Britons", and of the land as "Britannia" (which also included Anglo-Saxon and Norman England). The Latin self-identification they were moving towards was based on the new "Welsh" discourse initially derived from Anglo-Saxon writing (Old English "Wealh" or "Walh", meaning "slave", becoming "Walenses", "Wallia" in Latin). Gruffadd ap Cynan (King of Gywnedd) refers to Wales (the place) as "Gualis" in 1120. His son, Owain Gywnedd, who aspired to be King of Wales, styled himself "rex Wallie" and "Walliarum rex" (King of Wales/The Welsh). Meanwhile in the Eleventh Century the vernacular noun "Cymry" became well-established as a term for both Wales and the Welsh. However this term was also being used to refer to the Celtic people of Strathclyde, with whom the Welsh shared a language, so it's unclear how far this term really represents a departure from the old "Briton" discourse.

This answer relies heavily on Huw Pryce, British or Welsh? National Identity in Twelfth-Century Wales, which I strongly suggest reading if you have access to it, not least because it's an excellent source for further reading on Medieval Welsh writing.

edit: Pryce also offers an explanation for the shift in vocabulary: "It is likely that the Welsh discarded their old "British" vocabulary ... in response to changes in terminology on the part of Anglo-Norman writers." He also mentions the increase in Anglo-Norman and other Francophone settlers in Wales, and the phenomenon of Cambro-Latin writers being educated in England and being taught to use this new vocabulary there. He writes that "the main driving force ... was probably the need to communicate with the world beyond Wales".