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"I’ve been told that it’s best to pay the taxes on a Roth conversion out of an after-tax account so that all of the conversion funds can continue to grow tax-free. That would be my preference."

Yes, this is the point of "method 3" below, you still convert the entire amount, but in a way that includes withholding.

"I used TurboTax’s Annualized Income section, but I’m not sure I did it correctly."

I have never actually had to use the form in real life.  The form and instructions are here, if you want to try and follow IRS instructions instead of turbotax instructions.  https://www.irs.gov/forms-pubs/about-form-2210

The 4 "quarters" of the year are in fact uneven, the first quarter is 3 months, then 2 months, then 3 months, then 4 months.  The federal fiscal year starts October 1, so they want as much tax money by September 30.  Using 4 calendar quarters would put the 4th payment at October 15, in the new year instead of the old year. (I think that's the reason.)

"So in this case I’m still using converted funds to pay the tax, right? And would this then count as a $12K conversion?"

I can't tell if you understand.  Let me try to explain in a slightly different way.  Here are 3 ways to convert $10,000.

Method 1. Contact the traditional IRA and withdraw $10,000, they send you a check, you deposit it in your checking account.  You have 60 days to send the Roth IRA a check for $10,000.  If you do, that counts as a conversion of $10,000.  The Roth IRA needs to know this is a conversion (because the amount is too much for a contribution) but they don't do any checking to verify what you are doing.  You self-certify that it is a conversion, and less than 60 days has passed, and from then on any mistakes are between you and the IRS.  It doesn't even have to be the same money, just the same amount.  You could put the $10,000 in a savings account for 59 days and earn $80 of interest.  You could use the money as a down payment on a new car, and then sell your old car to raise back the $10,000.  As long as $10,000 goes into the Roth in 60 days or less from when you withdrew $10,000 from the traditional IRA, it is a completed rollover, and it doesn't matter if it is the same $10,000 or not.  

Separately of course, you pull $2000 from anywhere (same bank account, different bank account, under the mattress) and make an estimated tax payment.

Now, because income is assumed to be spread out over the year, but payments happen when they happen, the IRS sees $10,000 income spread out over the whole year and a $2000 lump sum payment late, so they assess a penalty because (in their mind) you should have paid $500 in April, $500 in June, $500 in September, and $500 in January.  You avoid the penalty by using the annualized income method on form 2210.

And if you miss the 60 day window to put the money in the Roth, then you have a taxable $10,000 withdrawal, and no way to put the money into any kind of Roth IRA.

Method 2. Contact the traditional IRA and perform a direct rollover of $10,000 to the Roth IRA without withholding.  This is your $10,000 conversion.  In many ways, this is more secure and better than an indirect rollover where a check comes to you, because it eliminates any risk of delay or mistake that would break the 60 day window.   Separately, you pull $2000 from somewhere, and make an estimated payment.  You have the same issue with the underpayment penalty and the need to use the annualized income method to minimize your penalty.

Method 3.  Contact the traditional IRA and perform a direct rollover of $10,000 to the Roth IRA with $2000 of withholding.  $8000 arrives at the Roth IRA.   If you stop here, you have an $8000 conversion.  But you go one more step.  Go to your bank account or your mattress, get the $2000 you have set aside for taxes, and instead, send that to the Roth IRA.  Tell them it is a conversion.  They don't need to know it is part of the larger conversion, and they don't care.  It doesn't have to be the same money.  As long as you get a total of $10,000 into the Roth IRA within 60 days of starting the process, it counts as a conversion of $10,000.  The IRS still has an additional $2000 of your money, but as withholding instead of as a payment.   

Remember that income is assumed to be spread out over the year, but payments happen when they happen.  Withholding is also assumed to be spread out over the year.  So the IRS sees $10,000 of income and $2000 of withholding, and they are happy, and don't assess a penalty for underpayment. 

Now, in your case you have an additional problem that, because you did not make estimated payments for your SE tax, the annualized income method will give you an underpayment penalty for the SE tax even if you don't owe an underpayment penalty on the conversion tax.  The trick to avoid this is to have a little extra withheld from the conversion, because -- withholding counts as if it was spread out over the whole year, even if the withholding occurred on one day late in the year because it was withholding from a  lump sum IRA withdrawal instead of spread out over many paychecks. 

Now the $21 is not too bad, and it may just be safer for you to make the conversion without withholding, make the separate tax payment, and use the annualized method.  I don't want to go over your head or make you uncomfortable.  But method 3 is an option to take advantage of the way that the IRS looks at withholding differently than payments.