When hear “locally owned” I usually think of mom-and-pop places, although some can be upscale. I’m not entirely sure how these things work, but a lot (most?) of upscale restaurants are owned by professional restaurateurs and backed by investors, etc.
The French Laundry probably isn’t considered a mom-and-pop place, but is it considered locally-owned ? I mean, Thomas Keller, the (sole?) owner lives in the area. But then what does that make his other restaurants, like Per Se in NYC? Is it neither a mom and pop or locally owned?
From what I can tell you’re referring to a restaurant that’s managed on it’s own, it’s just that someone (or some business) is backing it, right? So there’s only one of these restaurants (as opposed to a franchise).
I’d just call it a restaurant. I don’t think there’s any special term for it. You could call your’s locally owned but that’s a special case. I’d have no problem calling it locally owned since the owner does live in the area. People in the restaurant and business industry probably know him and would recognize him walking down the street.
As for restaurants where the owner lives hundreds of miles away and really doesn’t have much to do with the place other then a vested interest in making sure it does well enough that he/she gets their check each month, I’d call it a ‘restaurant.’