I was wondering if the coins in the Mint Sets are from the same run as the business strike coins or is there some distinction?
Different run, but that's all . . . same planchets & tooling. The only difference is the setup to make coinage for packaging, rather than bagging. There should be no difference in quality between coins destined for mint sets and for register drawers.
Once upon a time they were regular business strikes. From 2005 through 2010 the coins have a satin finish. More recent sets have coins that are described as brilliant and struck with extra pressure - I'm surprised the TPG'ers certify these as MS and not SP, as they are distinguishable from business strikes.
I think the 'extra pressure' sets actually began before the satin finish sets. I saw this somewhere, may have been in the insert from one of my late 90's - early 00's sets.
I think that the coins destined for the mint sets receive special handling to eliminate some of the marks which occur as the coins are shot into the ballista bags.
Checked last night and at least by 2003 the special presses with extra pressure were being used for the mint sets. However, well struck regular business strikes would be difficult if not impossible to distinguish from the mint set strikes.
Oh, I thought the only difference was that Mint Set coins were struck with new dies. After a few hundred (thousand?) strikes the dies were used for ordinary business strikes.