nb: According to the article, $530m is the notional value of his puts, not the premium he paid. 8k contracts is a fairly substantial trade on a $566 stock, but it's not as sensational as the headline makes it sound.
If the contracts were listed, option traders would look at big trades that hit the tape without contingent stock printing simultaneously, and call the brokers that crossed those trades to ask which bank sent the order to the floor. Then they'd compare that against banks which are believed to trade with Burry, and filter to get a guess at how much he traded. This may be tough with TSLA because there is so much activity, but in smaller names where only a couple of big prints go up daily, it's pretty easy to figure out who trades what. The banks that trade options against big players get quoted on a lot of stuff that eventually trades at a different bank, so they can often infer the identity of the client when they see the print hit the tape.
CNBC, on the other hand, know nothing and simply report what they are told and what they observe in regulatory filings. Their chief function is not to break news, but rather to distribute it -- like a buddy who is very up-to-date on current events.
TL;DR You can often deduce some of the characteristics of a position, but there are ways for a very stealthy market participant to hide the characteristics of his stake.