"Section 121 is an exclusion, not a deferral. So the basis is not affected by prior sales."

Nicely stated. It appears the second home sale doesn't meet the ownership or look-back requirements for exclusion, but it might meet "unforeseen circumstances" for partial exclusion.

The third home is a whole new situation, and their residency and ownership clock just restarted. Its basis stands alone. There is no carryover.

I believe the references here are what changed in 1997.