Do I have to pay my employer the full amount after an overpayment? ?

Taxed money was still paid to you by the employer, HMRC owe you a tax back.

Never in my life have I heard of an overpayment. Wouldn't they just take it out of future payments?

The over payment that was paid to YOU, you have to repay, the employer has to contact HMRC about any tax overpayment made although the tax will right itself next pay date

You repay 200 pounds.  Have your employer not pay the next 100 pounds of taxes and give that money to you instead.    

Of course you have to pay back the $200.

If they overpay you 200, yes, you owe them 200.

But if they took 100 tax on the 200, they owe you the 100 tax back.

The net of what you owe is 200 less the 100 tax = 100.