I live in LA and tried to take part in this gimme back in April...and got a ticket!
After much back and forth with the city, I was finally able to get it removed and get my money back, but it was a hassle for sure.
Also, note that if the program only lasted 8 months, it would have actually expired June 30, 2005, not 2006, so that's a typo in the article. It's hard to tell if the program is still around or not.
As for what qualifies as Los Angeles or not, good question.

It's hit or miss, as certain areas of the city incorporated as seperate entities years ago. Here's a partial list:
Burbank
San Fernando (the city, not the Valley itself, most of which IS part of the city of LA)
Glendale
West Hollywood
Beverly Hills
Culver City
And to be difficult, Marina Del Rey isn't its own city or in the city of LA--it's an unincorporated section of Los Angeles County (go figure).
A general rule of thumb: if you look up and see the standard LA blue street sign, you're in LA. Most of the special cities have their own signs indicating that you're in their city (White in Beverly Hills, Green in Culver City, etc.)
Hope that helps!