A few ideas on your Q
Your in a spot here. There are two cities in Sweden similar to your
1) Linköping in Östergötlands län
2) Lidköping in Älvsborgs län
but none like the combination "Lindkoping" or "Lindköping"
The other place name you mention could very well be Vinnerstad, (Old
spelling Winnerstad or Winnersta) located just outside Motala, a city on the
eastern shores of lake Vättern (you'll see the lake on any Sweden map)
Motala is only 44 km from Linköping so this seems more likely than the 2nd
However, a birth record would not read "Vinnerstad, Linköping" since it does
not match. If he was born in Vinnerstad the birth parish would be
Vinnerstad, not Linköping. If the notation you quote is not a birth record
then it could be a general localization or even "born in Vinnerstad, lives
in Linköping" Does any of this make sense ?
Dr Fredrik Haeffner