In the case of Tether, the company represented that each of its stablecoins were backed one-to-one by U.S. dollars in reserve. However, an investigation by the Office of the Attorney General (OAG) found that iFinex — the operator of Bitfinex — and Tether made false statements about the backing of the “tether” stablecoin, and about the movement of hundreds of millions of dollars between the two companies to cover up the truth about massive losses by Bitfinex.

also

Tether’s claims that its virtual currency was fully backed by U.S. dollars at all times was a lie.

The settlement also bars Tether from trading with New Yorkers, and the must submit regular reports about their compliance with this prohibition.

I suspect the AG got what they wanted, they just managed to do so without taking time in the court system. I would assume that Tether realized that it wasn't worth trying to defend themselves and so agreed to settle. My use of "successfully prosecuted" might not have been the right way to phrase it.

Reading between the lines - there was a period in the past where Tether was not fully backed. If Tether was still, as of today, not fully backed, would NYAG conclude this case?

I suspect the AG thought it had a bigger case involving ongoing fraud but settled as it could not find evidence of any ongoing false claims.

While biased in Tether's favor, worth a read: https://tether.to/en/stress-tests-resiliency-and-bank-runs/