I haven’t checked but how do they handle streets if the signs have two different spellings? There was a street growing up that was spelled two different ways and I still don’t know what the right one was!
My favourite example is a small road in Killarney called The Hahah in English but in Irish written "An Fhaiche" in one place and "An Háhá" in another. OpeStreetMap seems to designate the first one as name:ga and the second one as alt_name:ga.
This is a recurring problem in Ireland due to nationwide local government incompetence in using the Irish language.
Centuries old Irish language place names get replaced with bastard gaelicised versions of their English names, and now you’ve a mishmash of signage all over the place. Often the new names are just an invention of the council that sort of sounds right.
This is very true, but there is also an issue of which Irish orthography to use, right? Place names are very conservative, but modern speakers would be more used to O'Donnell's dictionary's variants than to ones from Dinneen, basically. Paradoxically, English forms sometimes give hints as to the correct pronunciation.
This tagging is most likely the result of a single OSM editor decision. They are quite often wrong / suboptimal / do not conform to guidelines.
It is always better to consult extensive OSM wiki to figure out how something should be tagged rather than try to figure out based on existing examples.
The rules are as open to change as the data I guess, but if there really isn't one obvious "proper" name I've seen things named with a slash like "name 1/name 2". Often there is specific tagging to overcome this, though. For example in bilingual countries like Wales many roads have any English and Welsh name. The language-specific names are uncontroversial, but the overall "name" (for the international map) is supposed to be in the "local language". As you can imagine this isn't always uncontroversial.