In Japanese, a mahou-tsukai 魔法使い is a "magic-user," literally, "someone who can use magic," mahou wo tsukaeru hito 魔法を使える人. Generally, however, mahou-tsukai is translated as "wizard."
A madoushi 魔道士 is pretty much the same thing, but since it's a different word, "sorcerer," I guess, who uses "sorcery," madou 魔道.
The term mahou-tsukai also refers to a virgin who's at least 30 years old. Or 25, in some cases.
That's because, according to the legends:
- sanjuusai made doutei dato mahou-tsukai ni nareru
30歳まで童貞だと魔法使いになれる
Virgin until 30 years old, able to become a wizard. - You can become a wizard if you're still a virgin at 30 years old.
For the record: that's a meme. A variant says you become a "fairy," as in, a spirit, yousei 妖精, instead. See: Tensei Shitara Slime Datta Ken for reference.
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