"My Life" by Bill Clinton, pages 381-2, 813, 854, 878-9, 907-8,934 (pages enumerated from Index which discuss the term "Third Way."

at pages 381-382:

I was amazed by some of the criticisms of the DLC from the Democratic left, who accused us of being closet Republicans, and from some members of the political press, who had comfortable little boxes marked "Democrat" and "Republican." When we didn't fit neatly in their ossified Democratic box, they said we didn't believe in anything. The proof was that we wanted to win national elections, something Democrats apparently weren't supposed to do.

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The principles and proposals I advocated could hardly be called Republican-lite or lacking in conviction. Instead, they helped to modernize the Democratic Party and later would be adopted by resurgent center-left parties all over the world, in what would be called "Third Way." Most important, the new ideas, when implemented, would prove to be good for America....

These are Bill Clinton's own words, but they could have easily enough been written by President Obama, who described himself as a "New Dem." I have no link, but heard him say this in an interview. Of course, I immediately recognized that term and knew instantaneously that I am much farther left than he.

I point this out to answer your question "... do the people you name self-identify as "Third Way"?

I believe you overstepped when you referred to the terms Third Way or DLC as "bogeyman terms." Obviously, these are legitimate terms used by people who want to be recognized as having separated themselves from -- now I will openly make up a term -- Classic Dems.

I have read several bios about Clinton, one of which specifically mentioned his search for a term to particularly identify his brand of politics. He wanted to differentiate his brand of politics from the DLC, from whence he came, thus the search for a different branding. He came up with the term Third Way, but I do not believe it truly stuck. New Dem in the political world today is synonymous with Third Way.

The DLC (Democratic Leadership Conference) openly publicly published its membership list, so there was no secrecy about it. I think of the New Dems as the "remains" of the DLC with a few having left and some new having joined.

Sam

Edited to add President Obama's declaration of being a New Democrat:

http://www.politico.com/news/stories/0309/19862.html

"Surrounded by 65 moderate Democrats on Tuesday in the State Dining Room, Obama was happy to portray himself as simpatico with a group of members who are largely socially liberal but fiscally more moderate to conservative."