Many people mention terms like "low" and "high" tenors. Are there formal definitions of "high tenor"?
A composer may indicate in the score that a role is intended for a "high tenor" (which does not mean a countertenor). An example is the role of Henry Morosus in Richard Strauss's opera Die schweigsame Frau (The Silent Woman). Another example is the role of the Italian Singer in his opera Der Rosenkavalier, who has but a single aria in the whole opera.
I took a quick look, and I counted six operas in which Strauss indicated a role (big or small) for high tenor and, although not counting, I noticed one role for low tenor.
The difficulty of high roles is not so much the highest notes required, but rather the "tessitura" of the entire part or aria. This refers to the range within which most notes of a vocal part fall. A role or aria can be impossible or pose risk to the voice if the tessitura is too high. The strain will usually be noticeable and a problem for the audience as well as the singer.
Transposing single arias is very often done in recitals (lieder are often transposed for different vocal ranges), and it's sometimes done for an individual aria in an opera, but rarely for an entire role. Massenet did an adjusted, baritone version of his Werther, originally for tenor.
But every voice type clearly has its limits. In his original, one-act version of Strauss' Ariadne auf Naxos, the showcase aria for the coloratura soprano role of Zerbinetta, "Großmächtige Prinzessin," went up to the F♯ above high C. Strauss took mercy (but barely a drop) when he revised the opera, adding a Prologue: he transposed this section of Zerbinetta's extended aria a half-step down, so it goes up "only" to F♮.
We use the German word fach for a voice type within a classification system (being very careful with our pronunciation). Our English system is a bit less elaborate than the German one. See voice type, which, referring to the Italian classification system, has:
Tenor subtypes: Tenors are often divided into different subcategories based on range, vocal color or timbre, the weight of the voice, and dexterity of the voice. Tenors are often broken down into eight subcategories: Tenorino, tenore contraltino, leggero tenor or tenore di grazia, lyric tenor, spinto tenor or tenore spinto, dramatic tenor, heldentenor, and baritenor.
I would definitely add "tessitura" to the list of qualities.
That would be the countertenor.