"There was a lower income threshold for the second stimulus payment."

Nope. Perhaps you are confusing Advanced payouts with actual rule. There is no change or split for the income eligibility.

"No bueno on the second round, but they should have received a stimulus check for the first round last March, but never did, so I am now trying to claim it."

"I knew they qualified for a partial first stimulus, but no second stimulus."

Nope.

Perhaps it would help to review what is really happening:

The funds were paid out as Advanced payment against a projection. The projection used 2018 or 2019 tax returns. But 2020 is the Actuals. You use the 2020 return to reconcile what a person is entitled to, against what they got.

If the person is not a dependent in 2020, then they would be entitled to the payment/credit as individual filers. That doesn't mean "not being claimed." It means "no longer qualifies as a dependent."

You might want to bookmark these links and read the IRS guidance.

Interactive wizards portal for determining dependency:

https://www.irs.gov/help/ita

And:

https://www.irs.gov/newsroom/economic-impact-payment-information-center-topic-a-eip-eligibility

https://www.irs.gov/coronavirus/second-eip-faqs#Eligibility

One for each EIP.

And here:

https://proconnect.intuit.com/community/federal-taxes/help/recovery-rebate-credit-for-tax-year-2020-...

It's something you will need to more thoroughly understand.