The stickers have varied somewhat over the years. They’ve been around for well over 40 years.
There’s “Central Hatzalah” for NYC and the Catskills. But each neighborhood in the city has its own division. Each division fundraises separately. And can print material on their own. But they all use the same emergency telephone numbers/call center.
The NYC Hatzalah is the original Hatzalah. It started in Williamsburg. Which is why the two phone numbers are both from Williamburg exchange (230- and 387-) numbers, originally from New York Telephone (now Verizon.)
Outside NYC/Catskills each Hatzalah is its own entity.