"Please confirm that my understanding here is correct"
Correct.
"But how does the new vendor know that?"
They don't, and it's not their responsibility to know that. It's your responsibility to report it correctly on your tax return. Because they know that you are over age 59½, they'll use code T rather than code Q on any Form 1099-R for a distribution from any particular Roth IRA that they do not know has met the 5-year qualification requirement. You'll need to tell TurboTax yourself that you had a Roth IRA long enough ago to satisfy that requirement so that TurboTax will treat code T the same as code Q so that no Form 8606 Part III is prepared. The IRS knows the year for which you first made a contribution to a Roth IRA because all contributions were reported to them on Forms 5498.
