Listed in the Roster of Union Soldiers, 1861-1865, as:

Colbertson, Walter NY 14th Hvy.Arty. Co.K

I have a hard time understanding this one. I don't doubt what you say, but if he were living in Kentucky at the time, why would he join a New York Heavy Artillery unit?
I would strongly suggest obtaining his military service records, either from the Bat'l Archives in Washington, DC or Broadfoot Publishing at http://www.broadfootpublishing.com.

I suggest this because this man may not be related. The unit was no where near Kentucky during the war.It organized at Rochester, NY and recruited from Monroe, Yates, St. Lawrence and Jefferson Counties between June 1863 and January 1864 while it did not muster into US Service for 3 years until between August 29, 1864 and January 17,1865. Even then it pulled garrison duty in New York Harbor until April 23, 1864 when it went to the front at Warrenton, VA where it joined 9th Corps and worked its way down to Petersburg, VA via the following battles: The Wilderness, Spotsylvania, North Anna, Bethesda Church, Cold Harbor, 1st assualt on Petersburg, mine explosion [The Crater], Weldon RR, Peebles' Farm, Fort Stedman, and the fall of Petersburg. Following the mine explosion at the crater, this unit was selected to lead the charge and was the first to plant their colors on the enemy works, capturing a Confederate flag in the process.
Mustered out August 26, 1865 at Washington, DC.

The Union Army, Vol II, p. 217

Union CULBERTSON's from KY are:
Culbertson, James A. KY 27th Inf. Co.E Sgt.
Culbertson, James E. KY 53rd Inf. Co.B
Culbertson, Robert A. KY 18th Inf. Co.H
Culbertson, William A. KY 30th Inf. Co.A
Culbertson, William W. (W.W.) KY 18th Inf. Co.K Capt.

Confederate CULBERTSON's:
Culbertson, Benjamin KY 3rd Cav. Co.A
Culbertson, Benjamin TN 9th (Ward's) Cav. Kirkpatrick's Co.I
Culbertson, Sam H. TN 24th Inf. Co.B
Culbertson, W.R. TN Arty.Corps Co.9 3rd Lt.
Culbertson, W.R. TN Lt.Arty. Morton's Co. 2nd Lt.