As I don’t speak English, my searches on the Forum are sometimes ineffective.
These messages help me to understand why there is this difference between the application and the website, and I understand the decision that was taken.
However, as Tiwan explained, the name displayed in the application is the name and its syntax in the iNat database :
In a collaborative approach, it would therefore be possible to make corrections directly in the database, in order to bring a little more consistency (even if it means debating, in a collaborative way, on a case-by-case basis).
As I’m involved in several collaborative projects (Wikidata, Wikipedia, OpenStreetMap), this kind of crowdsourcing seems realistic to me.