I have been tracing some of my tree that moved from Massachusetts to Vermont and found that several family members and relatives served in the Vermont militia under a Col. Williams [listed as 4 Reg't (Williams') Vermont Militia] during the War of 1812. There are also multiple people in my tree who I have moving from Lanesboro, Mass to Charlotte, Vermont around the same time, including one Col. William Williams.
Does anyone know if these are the same people? I will read the History of Eastern Vermont mentioned above, but thought I'd put the question out there.

The "History of Chittenden County, Vermont..." includes a bio of Col. William Williams and puts him coming to VT in 1788.

My primary interest is in forming a more complete picture of the wartime activities of the VT Militia's 4th Regt (under Col. Williams), as several people in this unit either migrated from Mass to Vermont together or then from Vermont to Huron Co. Ohio (REEDs, HUBBLE/HUBBELLs, BRECKENRIDGE/BREAKENRIDGEs).

Apologies in advance if these are not the same Colonels Williams.

Thanks!
J.R.