Why is "Puella Magi" equivalent to "Mahou Shoujo"? From my understanding, Mahou Shoujo = Magical Girl as the latter is normally used when turning the Japanese title to English, i.e.


Mahou Shoujo Lyrical Nanoha = Magical Girl Lyrical Nanoha
Sasami: Mahō Shōjo Club = Sasami: Magical Girls Club
Mahou Shoujo Lalabel = Lalabel the Magic Girl


Now, I understand one of the themes of Puella Magi Madoka Magica is Magical Girl, and its Japanese name is Mahou Shoujo Madoka Magica. I am wondering how did Mahou Shoujo become Puella Magi?
The romanized title is not meant to be an English translation at all. It is not half Latin + half English; it is all Latin from start to finish: Puella Magi Madoka Magica. So we shouldn't read the "Madoka Magica" as if it were English and then wonder at why "Puella Magi" is not English. "Madoka" is the main character's name (鹿目 まどか), so that word is a native Japanese word. "Magica" is clearly not Japanese or English; it is also Latin. So 3 out of the 4 words are Latin, and the Japanese word is a person's name, so it would naturally be written "Madoka" in romanized letters when in an all-Latin phrase.

According to the Puella Magi wiki, "Puella Magi" in Latin translates to "girl [form] of the sorcerer" in English. As this was the translation decided by the Japanese company, it is possible that they did not consult anyone fluent in Latin in order to translate "mahou shoujo" as accurately as possible (we often see Engrish in anime [such as 「それでも世界は美しい」(Soredemo Sekai ha Utsukushii), which was officially written in English as "Still world is Beautiful" instead of a literal fluent translation like "Even So, The World is Beautiful"]; there is no reason that they would necessarily do better with Latin).

The wiki author makes an alternate hypothesis, saying "If interpreted a different way, 'the sorcerer's girl', it implies that the girls are being used - which they are, by Kyubey," but since we do know the meaning of "mahou shoujo" due to its use in the many, many magical girls series that Madoka is parodying, we can conclude that it does not in this case have a unique meaning of girls being used. The series is not a mahou shoujo series proper (it is a parody of the genre and is targeted at male viewers, whereas real mahou shoujo are a subset within shoujo, meaning they are targeted at an audience of young girls and if there is a manga origin or adaption, it runs in a shoujo manga magazine), so for it to use the standard style and meaning of a mahou shoujo series best serves its intent at parody.