When and why did a set of NY theaters become "Broadway Theaters?" I've
found, between Wikipedia and the livebroadway.com site, some sense of
what these theaters are now, but not the history of when they became a
distinct entity and what led to that.

Hello aeisaman44-ga,

I found a wonderful website http://www.talkinbroadway.com that tells
exactly what you want to know.

It says between during the 1900-1910 was the transition from "a area
of theater" to Broadway.

"In the 1900-1901 season there were seventy plays or musicals being
produced on Broadway. It was the beginning of the boom, and the
decades that followed saw that number quadruple. In addition, there
were seven Vaudeville houses and six Burlesque theaters presenting
their shows to a theater thirsty population of just over three and a
half-million inhabitants."

if you read the page on the 1900-1910 decade you can see many famous
actors, plays, and musicals were produced during that area and it GREW
in size. The addition of "Times Square" as well made it more than just
a district. It also had a transition from more Victorian and European
based acting styles to the more American realism style associated with
Broadway.

Google Searches Used:
://www.google.com/search?num=100&hl=en&lr=&safe=off&q=history+of+Broadway&btnG=Search
History of Broadway

If this answer requires further explanation, please request
clarification before rating it, and I'll be happy to look into this
further.

Nenna-GA

This page may be of interest to you:

http://www.michaelminn.net/newyork/index.php?broadway_theatres