Do I file amended returns, or just deduct the refund of prior years' medical expenses I just received from my 2018 medical expenses when I file my 2018 tax returns?

Neither.  If you received a reimbursement deducted on a previous tax return, that refund is still  taxable income.  It goes on 1040 line 21 as Other Income.
Federal Taxes
Wages and Income
Scroll way down to the end  - Less Common Income
Very last one….Miscellaneous Income - Click start or Update
Then it goes on the second line for Reimbursed Deductions from a prior year

Were you able to deduct some medical on the prior year return?  You could only deduct the amount over 7.5% of your AGI.  So it’s limited, so I don’t know how you account for that.

You did PAY (not just incur) those expenses in 2017, right?

Yes.  I paid them in 2017, and I'm on the cash basis.

Yes, my medical expenses in 2017 exceeded 7.5% of my AGI, so I was able to deduct the difference.

So you don’t have to claim all the reimbursement, but we’re not sure how Turbo Tax will handle it.  It may do it for you.

I understand now that the refund I got should is considered to be an itemized deduction recovery, and that I must count it as Other Income to the extent I received a tax benefit for the deduction in 2017 and in all applicable prior years.

You don't do either of the things that you suggested. The refund that you received is called an itemized deduction recovery. You have to report it as income on your 2018 tax return to the extent that you received a tax benefit for the deduction in 2017. TurboTax will not do the calculation for you to determine how much of the refund has to be reported as income. If subtracting the amount that was refunded would have brought your medical expenses below 7.5% of your 2017 AGI, you don't have to report the full amount of the refund as income. Otherwise you probably do have to report the full amount as income.

If you have to calculate the taxable amount, follow the instructions for itemized deduction recoveries in IRS Publication 525, and enter the taxable amount that you calculated in TurboTax. If the amount is not large and you don't want to bother doing the calculation, you can just treat the full amount as taxable. That is the most likely result anyway.

Here's how to enter itemized deduction recoveries (other than state or local income tax refunds) in TurboTax. These instructions are subject to change since the 2018 TurboTax software is not available yet, but the procedure has been the same for many years.

• Click the Federal Taxes tab. (In TurboTax Self-Employed or Home & Business click the Personal Tab.)
• Click Wages & Income. (In TurboTax Self-Employed or Home & Business click Personal Income.)
• Click "I'll choose what I work on" or "Jump to a full list."
• On the screen "Your 2018 Income Summary," scroll all the way down to the last section, "Less Common Income."
• Click the Start, Update, or Revisit button for the last topic, "Miscellaneous Income, 1099-A, 1099-C."
• On the next screen, "Let's Work on Any Miscellaneous Income," click the Start, Update, or Revisit button for "Reimbursed deductions from a prior year."
• On the next screen, "Reimbursed Deductions," put a check in the box for the appropriate type of deduction and click Continue. You would check the box for Medical Expenses.
• On the next screen enter the taxable amount of the recovery that you calculated. The screen says "Enter the total reimbursements you received . . .," but that is incorrect. If the taxable amount is less than the full amount that was refunded to you, enter only the taxable amount. Click Continue.
• Click Continue again to return to "Your 2018 Income Summary."

The taxable amount of the recovery will be reported as "other income" on your tax return with a description that depends on which box you checked for the type of deduction. Exactly what form and line it will appear on is not certain yet, because the IRS is still working on the design of new tax forms for 2018.

So this TT FAQ is wrong?
<a rel="nofollow" target="_blank" href="https://ttlc.intuit.com/questions/1901115-do-i-need-to-report-medical-expense-reimbursements">https://ttlc.intuit.com/questions/1901115-do-i-need-to-report-medical-expense-reimbursements</a>

as are these?
<a rel="nofollow" target="_blank" href="https://ttlc.intuit.com/questions/3029119-received-the-reimbursement-of-medical-expense">https://ttlc.intuit.com/questions/3029119-received-the-reimbursement-of-medical-expense</a>
<a rel="nofollow" target="_blank" href="https://ttlc.intuit.com/questions/3267856-how-do-i-report-medical-reimbursement-income">https://ttlc.intuit.com/questions/3267856-how-do-i-report-medical-reimbursement-income</a>

Those links all say essentially what I said, with somewhat less detail. What exactly do you think they say that's wrong?