In English, it’s incorrect to use Common Hackberry, the correct form is common hackberry. Used in captions or at the beginning of the sentence, I recommend to apply only sentence casing, but not title casing, to the common names provided. Basically take the name string provided under the names section for the given language, and turn the initial letter to capitalized, if it isn’t already.
You say “it’s incorrect” but there is no absolute consensus on what is “right” or “wrong” in English regarding using Title Case for common names. A couple of centuries ago that was the norm, and there has been a gradual drift towards using lower case ever since, but that long slow march is by no means complete. There is still a significant minority of English-language publishers that continue to use Title Case for common names, especially in ornithology, where Title Case remains dominant. Indeed for years the Wikipedia editorial style guide said to use Title Case for common names of birds (but lower case for the majority of other life groups), although that rule has recently changed to a more universal lower case rule.
Using Title Case for common names is necessary to avoid confusion… was it a blue butterfly (such as a Blue Morpho) or a Blue Butterfly (Polyommatinae) that you saw? Was that a checkered white butterfly (such as a Western White butterfly) that you saw, or a Checkered White butterfly (Pontia protodice)? Was is a gray hairstreak or a Gray Hairstreak? Was it a giant swallowtail or a Giant Swallowtail?
Just because most people are doing it wrong doesn’t mean that you should also do it wrong. That’s what sheep do, they follow mindlessly. We are humans, we can do the right thing even when most people don’t. Be the change you wish to see in the world – use Title Case for common names! And while you’re at it, also use the Oxford (serial) comma.
I don’t think this is how common names work, and may I say: what you describe is not how identification at iNat works. Languages have their own rules when it comes to common names’ capitalization, and we’re not in the position to overrule that.
As I like to say half-jokingly, English has no rules. And in the case of capitalizing common names, I prefer to do so when the name is largely standardized. It often serves to avoid confusion. I think the birders have the right idea. Some prefer not to. So it goes.