That's not entirely correct.
Like I said, depends on your insurance company, and perhaps state law. Obviously your own insurance company is going to subrogate, and work to get their money back from the at-fault driver. If you stay with that insurance company, perhaps your rates stay the same, maybe not, but they're not going to go into great detail if and when your rates do increase (the standard reason these days is inflation).
But what most people don't understand is that nearly all claims (at fault or not at fault) are reported to a claims history database, which is usually CLUE by LexisNexis. Even LexisNexis claims on their website [
https://risk.lexisnexis.com/products/clue-auto] that 99% of insurers report their claim data to the database.
If you happen to shop around for a new insurance company, that claim you made with your insurance is going to be in the CLUE database. It will show as zero dollars paid out (not at fault), but there's a claim in there. Most insurance companies will ding you for that when you are shopping new policies, even though you were not at fault. Do some research, I'm not making this up.
On the other hand, if you work with the at-fault drivers insurance company, the only claim that's opened is with the at-fault driver.
If you're one of the people that stick with State-Farm, Liberty Mutual, whoever, for 20+ years and have an established relationship and don't plan on changing, this probably isn't something you need to worry about. But if you shop for new insurance companies routinely to find the best rate, it's best not to open up claims in your name.