A comparison of the signatures of David Lindsey of Westmoreland Co., Pennsylvania  and
David Lindsay of Harrison Co., Kentucky  shows that they were not the same person.
 

This is the 1772 signature of  David Lindsey of Tyrone Township, Bedford Co. (later Westmoreland, then Fayette), PA.  David signed a petition  requesting  that Rev. David McClure minister to the local settlers.
This is the 1788 signature of  David Lindsay of Bourbon Co.(later Harrison), KY.  David signed a petition requesting that a tobacco warehouse be erected at Ruddle's Mill.