You're right, the public background will be accessible to people asking on the street in your character's land of origin. So anything that those people would not know should be in your private background!
In-game, you'll have the ability to annually update backgrounds for whichever land you're in currently. That will be useful when it comes to spreading your characters' reknown. But if necessary, game admin can also review your character application and notice that maybe your character should start out with a public background in a different area, as well.
So, your questions about the boundaries of distance are very apt. The lands of origin (in character background terms) are wider than single provinces, so it would definitely be possible for a VNPC from Omrazir to be chattering about a well-known Zalawi merchant (for example), because all of that would be happening in roughly the same continent. Someone from Rahoum, despite being in a different continent from Omrazir, would probably have some background that could be researchable by anyone anywhere in the Sirdabi Caliphate. You can be pretty generous in that regard. But if your character had been north of the Adelantean Sea and gotten into criminal trouble in Marpagna, it's unlikely that people in Omrazir would automatically know about that.
I'll address each of your examples to try to be more clear, but yeah it's not something we need to be very strict about, I think. It all has to do with how you want the public perception of your character to be, because the public background details aren't just rumors that a person could dig up by talking to your personal acquaintances -- they're public knowledge that any random VNPC could tell anybody.
He was a famous breeder of fighting chinchillas, known among criminally-inclined natives of Sirdab.
This could or could not be something that you include in your background. If your character tries to hide their involvement in the black market of chinchilla fighting, then it's probably better for your private background only.
In Zalawi, his place of Origin, he is known under a different name for being the chess genius son of a court advisor who apparently drowned at the age of 17.
Because the 'place of origin' factor is as broad as continents and empires (in this case, the Sirdabi Caliphate), that would probably be something publicly known if indeed it was publicly known. I think if he apparently drowned at age 17, it'd take someone who really knew him to recognize him by that different name... like maybe a strategically placed NPC who could throw an awkward wrench in his plans to infiltrate the Zalawi court! Again, up to you, but I would think this specific detail would be for a private background rather than public.
In Zalawi, known to underground Shamans but not to the general public, he was spirited away to learn to manipulate the mysteries of the universe.
This would be a private background thing for sure because it's not for the general public.
In Omrazir, there is a catering business that serves the rich and famous (no scrubs) owned by a man who fits his description but goes by another name.
I think it depends on how you plan to play him. If you plan to mostly play him in the role of this man, it's a good idea to have that as his public background. But if you're intending to majorly play him in the persona of someone else (his real self?) then it'd be a good idea for this to be a private background element.
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In general, it's completely up to you! If this character is so nebulous and secretive that it's genuinely difficult to pin down any single background to present as a major persona, then it's completely legitimate to write something either edgy and mysterious like "Nobody seems to know who this guy really is! What a mysterious fellow..." or "He seems to go through life like pretty much anybody, with family back home in Zalawi, and is known by most people as an affable and quiet person." It's up to you in terms of whatever you think is appropriate and how you'd like your character to be publicly presented.
To reiterate, anyone from anywhere in the Sirdabi Caliphate as well as most locations in Idiri would be able to find out the public background of someone from Zalawi.
Hope that helps!
