What is the true etymology of "algebra"?

It comes from (part of) the name of the fellow, Al-jebra, who first wrote out the method of balancing mathematical terms. The use of balancing then went to medical usage for bone setting. Others may have done it first but getting into print in 1550s is tough to beat.

Here is one source but I have seen plenty more all about the same. "formal mathematics; the analysis of equations; the art of reasoning about quantitative relations by the aid of a compact and highly systematized notation," 1550s, from Medieval Latin algebra, from Arabic "al-mukhtasar fi hisab al-jabr wa al-muqabala" ("the compendium on calculation by restoring and balancing"), the title of the famous 9c. treatise on equations by Baghdad mathematician Abu Ja'far Muhammad ibn Musa al-Khwarizmi

etymonline.com/search?q=algebra

The answer in my book, Mathematics A Curious History p.96 by Joel Levy confirms your general understanding.

The word "algebra" comes from the title of a book by the medieval Arab mathematician Al-Kitab-mukhtasar fi hisab al-have w'al-muqabal, which was written around 825CE - the Arabic word al-jabr became "aljebra". The title is usually translated as "The Compendious Book on Calculation by Completion and Balancing", although the last phrase can also be translated as "Reunion and Opposition".

In other words, according to Levy, al-jabr meanscompletion or reunion. He goes on to explain:-

<...the book> gives step-by-step instructions about how to solve algebraic problems through the two steps mentioned in the title: reunion (or completion) and opposition (or balancing), today known as transposition and cancellation.

To trace the etymology of al-jabr further is interesting, but goes beyond the scope English language usage: The name of the method of finding unknowns by the manipulation of equations came from the name of al-Kwarizmi's book.

On the other hand, the actual algebraic method goes back much earlier. To find the area of a field in the Nile delta with a view to taxing it required carrying out something like the instruction

multiply the length by the breadth to find the size/area.

To all intents and purposes, this was algebra Egypt-style. They even found a complicated way of calculating a field in a semicircular bend in the Nile. So at least a shadow of π was already there, leading by the time of Eratosthenes a pretty respectable stab at calculating the circumference of Earth.