Yes, the language is a bit confusing, so I understand why you are a little into the weeds on this.
It’s a sign of performance–the API Certification Mark, or as it’s commonly called, the “Starburst.” The mark is familiar to many in our industry–but which oils get marked? What performance standards does the mark confirm? How are these standards defined? And developed?
In simple terms, the mark means that a given lubricant meets current fuel economy and engine protection requirements defined within the current International Lubricant Standardization and Approval Committee (ILSAC) GF oil specification.
So the Starburst, as I noted, denotes the ILSAC GF-xx compliance.
And then we have, as @kschachn
noted, the API Service Symbol, which denotes API SN/SN Plus/SP...etc and can be for Gasoline (API Sx), Diesel (API Cx) or both.
You are right that it sounds like Honda's language implies the ILSAC Starburst, but I expect they indicate an API service category there somewhere. Mobil is of course on the board of the API, so their interpretation here is a bit interesting.
In either instance, the oil is approved by the API, but Euro lubes are often not ILSAC certified.