Acknowledging the debt can, in some states only, restart the limitations period. That is, the period during which you can be sued. A mere verbal acknowledgment is insufficient in most states.
Acknowledging the debt can NEVER restart the 7-1/2 year credit reporting period.
Even in states where acknowledging the debt restarts the limitations period, it does not create a new debt. It the the same 20 year old debt.
I still think that depends on how devious they are and how gullible you are. If you send them an IOU for $100 which doesn’t reference the original debt (the slight of hand I talked about earlier) then you are indeed responsible for that IOU.