Just as in Ukrainian, an English toponym very often consists of a specific paired with a generic. It’s also acceptable to omit the generic when the context is abundantly clear, but name=* doesn’t necessarily reflect this informal shortened name.
The feature in question is a nature preserve surrounding a ridge that’s known as Coyote Ridge in English. Because parks are human organizational constructs, we pay a little more attention to their official names than for natural features. The reserve’s owner made the decision to officially rename the preserve from “Coyote Ridge Open Space Preserve” to “Máyyan 'Ooyákma – Coyote Ridge Open Space Preserve”, redundantly prepending the Chochenyo name of the ridge. The ridge is still Coyote Ridge in English and Máyyan 'Ooyákma in Chochenyo.
One could argue that this is only the official_name:en=*, and that common usage still favors keeping “Coyote Ridge Open Space Preserve” as the name:en=*. However, the people involved with the preserve included the Cochenyo name very intentionally, and I don’t know of any controversy regarding that change. For a map that otherwise has all the formal names, selectively omitting the Cochenyo part of the English name would surprise visitors as much as truncating the name to “Coyote Ridge” on the map. “Coyote Ridge Open Space Preserve” is currently tagged as alt_name:en=* rather than old_name:en=*, out of recognition that people will sometimes skip the parts of a name that they don’t know how to pronounce or type.
Sometimes we do simplify the name for practical reasons. I’ve been tempted to shorten “The Heekin Family/PNC Grow Up Great Adventure Playground” to just “Great Adventure Playground” based on what people have been shortening it to online, despite the signs, but it’s really a gray area. I’m not sure everyone who passes by even knows whether it’s “Great Adventure Playground” or just “Adventure Playground” after removing the names of the sponsors (the Heekin Family and PNC’s Grow Up Great initiative). A lot of parks and other recreational facilities are like this.
Right, and this is one end of a spectrum. On the other end, many English speakers write “Huế” to avoid confusion with “hue”. In the middle, places can take a variety of forms, like Đà Nẵng (Da Nang, Danang), Điện Biên Phủ (Dien Bien Phu, Dienbienphu, Dienbien Phu), and even Vietnam (known as Viet Nam in diplomatic contexts). Anglicization is a process, not an event. These names at every step of the process are all English names by virtue of being used in English.
You want a clear criterion, rather than a preference, but your preference for a clear criterion is at odds with the local English-speaking population. As I said, English liberally borrows from other languages, not always very cleanly. Because the English language is almost completely unregulated, Anglicization is neither deterministic nor rule-based, nor is de-Anglicization. The transformation codes you’re proposing presuppose a clear distinction between pure and impure English that doesn’t generally exist in practice. I do not deny the existence of such a phenomenon in Ukrainian, however.