In my notes on the "linguistic" definition of salah, I have a few different points. Even though Shaykh Yaser Birjas in Fiqh of Salah discussed the linguistic root of the word salah, and so did Shaykh AbdulNasir Jangda in Vocabulary of Salah, the only summary I can give on the subject is that it's not immediately obvious what the root is, or if the word came from another language.
However, there are still some linguistic understandings inherent in the usage of the word, and that is what I will share inshaaAllaah.
So the first linguistic understanding of the word is that it means du'a and istighfar, which in English means supplication and seeking forgiveness. Du'a, or invoking Allah, is the essence of prayer and worship. In English we use the word "prayer" to mean both du'a and istighfar (and even dhikr) but these words in Arabic have special connotations. We can understand however that salah means invoking Allah and asking for His forgiveness.
The second linguistic understanding of the word is that it means forgiveness and mercy. We get this interpretation, which applies to the salat of Allah, from the following ayah:
Allah sends His Salat (Graces, Honours, Blessings, Mercy, etc.) on the Prophet (Muhammad SAW) and also His angels too (ask Allah to bless and forgive him). O you who believe! Send your Salat on (ask Allah to bless) him (Muhammad SAW), and (you should) greet (salute) him with the Islamic way of greeting (salutation i.e. AsSalamu 'Alaikum). 33:56
In this translation (Muhsin Khan), the word Salat is taken to mean "Graces, Honours, Blessings, Mercy, etc." when referring to the Salat of Allah. And then the Salat of the angels is asking for Allah to bless and forgive [Muhammad, saws.] And our salat in this case means asking Allah to bless him.
The way Sh. AbdulNasir explained it was to say that if Allah is the "doer" then the salat means to have mercy on, and to grant blessings. And if the "doers" are the angels then it means that they seek forgiveness. And if the "doer" is a human being, then it simply means to worship.
One other meaning of the word salah, is that the plural form (salawat) can mean masajid, or places of prayer, as in ayah 22:40.
The technical definition of salah, however, is how we understand the term today. That is, worshiping Allah by means of particular prescribed sayings and actions in a particular order at particular prescribed times, beginning with takbir (Allaahu Akbar) and ending with tasleem (As-salaamu alaykum).