I wish we could get past this recurring theme of titles. No - web designers are not calling themselves UX designers, any more than UX designers are calling themselves web designers - I don't believe what you have seen can be a generalisation so much as a few occurrences of note.
I think the field of UX is still relatively young and evolving - so not had time to really settle and as such there is still this confusion over terms, titles and labels.
I heard Dan Rubin (Creative Director at Moo.com) say recently that when looking at a list of average salaries in London for a person in this field, the list of roles consist of around 30 different titles all for exactly the same thing doing the same job but with a £10,000 difference between the average salaries.
Some people even have a different title internal to a company and another as seen by clients outside the company!
I think it's important not to focus on the title, but on the responsibilities: Treat the title only with the pinch of salt it deserves.