Picture a child leaving the house to be taught somewhere else, with other people's children, instead of being taught at home by a tutor hired for the family. That leaving-the-house is what public first pointed at: not the family's own arrangement, but a shared one. Everything else the phrase does today comes from arguing about who the sharing is with, and who pays for it.
From public + school: originally public in the sense of communal, in contrast to a private tutor; later public came to mean state-run, in contrast to private schools.
由 public 加 school 組成。最早的 public 指的是「共用的」，相對於自家請的家教。後來 public 的重心移到「政府辦的」，才變成和私立學校相對。英國那一支的名字停在舊意思，多數國家那一支跟著新意思走，於是同一個詞分成兩邊。
Wiktionary, entry 'public school' (noun), etymology note, source_ref kaikki:3f6f69a190036ae4:328392716
Two branches
政府出錢、人人可讀
共用的範圍擴到最大：全社會一起出錢，住在這區的孩子都能進來。
noun · B1
A school that is paid for and run by the government and is free for local children to attend. This is the usual meaning in the United States, Canada, Australia, New Zealand and many other places, and it was also the older meaning in Scotland.
公立學校
英國：收費的老牌寄宿學校
共用的範圍停在比較小的地方：收的是外面來的學生，但仍然要付錢才進得去。
noun · B1
In Britain and Ireland, an old private school that parents pay for, usually for students aged about 13 to 18, and often one where the students live at the school during term.
（英國）收費的私立寄宿學校
以下是外部詞典記錄的用法，原文照列，我們還沒有寫成教學內容。上面的說明才是我們自己寫的；有些用法可能上面已經講過了。
public schoolnoun
Certain private schools, particularly (initially) any grammar school operated apart from the personal lands of its students or (from the 19th century) the feepaying secondary schools which developed from or were modelled upon them; a British boarding school.
A publicly funded and administered school; (UK, Ireland) such schools in the context of other countries.
obsoleteA college or university.
資料來源：英文維基詞典 English Wiktionary（CC BY-SA 4.0）。 原文照列，未經改寫。