Spartans didn't call themselves Spartans. Their own name for their nation was Lacedaemon. (Or Lakedaimon, spelling being variant in these matters.) A Spartan was a Lacedaemonian. There were also the short forms Laconia and Laconian. That's why Spartan shields had the letter lambda (Λ) painted on them.
I prefer to write Spartan rather than Lacedaemonian in my books, and I'm pretty sure you prefer to read Spartan. But there's an interesting consequence of them being Laconian.
The Laconians had a reputaion for being men of few words. That's the origin of our word laconic. When we call someone laconic today, we're saying that they're as short-spoken as a Spartan.
The most famous laconic statement of all occurred at the Thermopylae, where 300 Spartans held for 3 days against an army of 100,000. (No, I'm not exaggerating the Persian side.) The Spartans were warned that the enemy was so numerous that their arrows would blot out the sun, to which one soldier named Dienekes replied this was good, because, "Then we will fight in the shade."
A similar situation arose when Philip II of Macedon (the father of Alexander) sent a message to Sparta suggesting they submit to him, because, "If I win a war against you, I will enslave you all." Sparta sent back a single word reply: If
Philip decided to give Sparta a miss.
The Spartan characters who appear in Sacred Games are not laconic. There are several reasons for this, first being that a book in which half the characters speak in mono-syllables is not exactly a positive.
The second reason is that laconic Laconians must be the exception if they wanted to run any form of society, and then there's the natural variation of personality. Not all Italians gesticulate when they speak!
Surviving examples of laconic speech aren't everyday speech; they're all pithy statements designed to hammer home a point. And that, I suspect, is the origin of the laconic Laconian: when they wanted to make a point clearly known, it was just a cultural thing that they did it with a short, powerful statement.
I very much doubt they were as dour as the laconic reputation suggests for this reason too: that among the Greeks they were known as "crickets" as a nickname, because the Spartans were always ready for a song and a community dance. That doesn't say laconic to me.