Disclaimer: I'm not an opera singer or a professional singer by any means. Just a hobbyist of over a decade, and I just happen to have spent a lot of time researching the subject of high tenors.
Within vocal types like Tenor, Bass, Alto etc... There are sub-types, in opera they are usually referred to as Fachs. Though in contemporary music they are usually referred to as "High/Low Tenor", "High/Low Bass", in choral music it's "Tenor 1/Tenor2"... There's a lot of variances in language.
But generally speaking, when someone says high tenor they're probably referring to one of the higher Tenor fachs, one of the highest being the Leggiero Tenor. Interesting resource on that type below
Other sub-types can be found here: https://en.wikipedia.org/wiki/Tenor#Subtypes_and_roles_in_opera
In terms of defining your "fach", it is less about your range and more about your tone throughout your range. Plenty of high tenors can sing below C3, but there is usually a sort-of change in quality around E3 and below. Usually a decrease in resonance. Sometimes the voice gets breathier.
And then there's the "passagio" or break, for a high tenor (lyric or leggerio) will be around D4-E4. Here and above, there will again be a change in the quality of the voice. This is important for determining Fach, but it is also important to keep in mind that Fach isn't all that important unless you're singing opera. In a way, you don't really have a "fach" unless you do sing opera and are trained to a certain level. Because the fach system is built upon the standards that opera singers are held to (in terms of technique). There isn't really a technical standard in pop/contemporary music, there's all sorts of techniques and philosophies, and many singers just sing without thinking much about technique. So, if you're a pop singer with a high tenor range/voice, it might be more accurate to say that you're a high tenor who would probably be a lyric/leggerio tenor in opera.
I could ramble on and on. Love talking about this stuff so let me know if you have any questions.