Although it is impossible to know for sure about Elizabeth Flatt, in the DNA research done so far, Vliets and Flatts are clearly two separate lines and not related. We have now tested kin of Flatts who were in New Jersey, New York, Pennsylvania, Tennessee, Kentucky, Illinois, etc, as well as Vliets who are descendants of Dirck Jansen Van der Vliet. http://www.halcyondays.com/flatt/flattlinesdnaall.asp
The Vliets, as you can see, are of the E1b1b1 haplogroup and clearly kin. Flatts of Norfolk UK are of the I2a Haplogroup and clearly kin. The Flatts who are descendants of John Flatt b 1759 NJ, Andrew Flatt b 1755 NJ, John Flatt m Mary Savage, Benjamin Flatt b 1776 PA, and Alexander Flett b 1884 Erie Co NY are of the R1a1 Haplogroup and clearly kin. Most remarkable in that last group is the relationship with the Flett of Orkney Islands, Scotland, which is also R1a1 haplogroup.
At this point, there is no DNA evidence that the Flatt is a corruption of the Vliet surname. Both Vliets and Flatts lived in the same area of NJ, for example. And definitely descendants of John Flatt b 1759 NJ who married P E Logston/Logsdon are not descendants of Dirck Jansen Van Der Vliet.