If you think it's terrible for communicating to the average person (and I am very much the average person) you could explain it yourself in detail.
So I will try:
You are right that it is a sort of consensus rating threshold maintained by the ratings agencies that means their (Anthropic's) debts are not considered investment-grade — that they are speculative, high risk, that they are considered likely not to be able to repay. It is ultimately the ratings' companies (and therefore some individual people's) decisions as to whether they are or are not junk-rated.
But for Anthropic it also means they will pay much more for their debt, they may not be able to negotiate long repayment terms, they may need to de-risk with underwriters by paying higher fees, they may be limited by the terms of any individual loan — it may limit how much future debt they are allowed to take on, may mean they need the lender's approval to restructure their business in even small ways, etc., and most importantly, some institutional investors (pension funds, mutuals, insurance companies, ETFs etc.) will be barred from investing (which drives why their debt will cost more).
Frankly I think it is baffling that Anthropic are still headed to an IPO when this is what it could look like on the other side.
Previous bubble villain Michael Milken and the 80s leveraged buyout bubble made it a name of common parlance because that bubble was built on "junk" bonds.
The old traders of that time who were against what Milken was doing called them "junk" and the name has stuck ever since.
It just means "risky". If you trade bonds it has no association with "bad". It is an inside bond market term, I think that is why it sticks too. It is a fun term if you trade bonds but terrible for communicating to the average person.