I received a second stimulus payment but not until January 2021. Do I still add it to what I'm claiming for 2020 taxes?

Yes, you will record the $600 you received in January as the second stimulus payment. 

1. Log into turbo tax
2. Click on Federal Review 
3. Let's make sure you got the right stimulus amount>continue
4. Next screen asks Did you get a stimulus payment? Yes
5. Then a drop down will appear and this is where you enter your first and second round stimulus payments.

Why is this recorded against 2020 taxes when it wasn't received until 2021?

You are not being taxed on this, it is just being reconciled with your 2020 return, regardless of when it was received.  If you didn't receive your full payment, it is being added to your refund,

So by the looks of it, even though we're not getting taxed on these amounts, we're really having to pay back the amount of the stimulus received.  Am I seeing this right? If my mom hadn't received these stimulus, she would only have to pay $377.00 instead of $2177.00.  So these were a "loan."

No, you are paying back the stimulus.

When you start preparing your return, TurboTax assumes that you haven't received  the stimulus payments, so it adds the stimulus payments to your refund.

Once you enter the amount of stimulus that you've actually received, it adjusts your refund (amount due) to account for that.