I just discovered the term "Anno Hegirae" for the first time (I'm not Muslim).
I'm curious, why does the term use Latin rather than Arabic? Is there a version of the term in Arabic? What is that term (in Arabic and transcribed into English).
The proper Arabic term is من الهجرة (min al hijra), which literally translates as "from the time of the Hijra". So right now it's 1435 من الهجرة, usually shortened to 1435 هـ.
However, in English the term 1435 AH is usually used, which is (as you have discovered) short for anno hegirae. This is a modern usage that deliberately corresponds to the usage of AD (anno dominee) for the Gregorian calendar. I have no information on when this usage was adopted.