Folks,
Here is a good reference on the chronology of the Spencer Rifle/Carbine
http://www.civilwarguns.com/spencer1.html
The action was copyrighted in 1861, and the first Federal order was for 700 rifles for Navy use (June 1861). An order was placed for 10,000 rifles for Army use in December of the same year. Because of manufacturing problems (Spencer did not have a factory/armory at the time of the orders) the Navy repeaters were not delivered until February 1863, and the Army order was actually bumped up, with the first being delivered in December 1862. The first unit issued these rifles was the 5th MI
Cav, and the order was eventually reduced to 7,500 rifles.
Contrary to popular belief, the Spencer Carbine was not present at Gettysburg and did not have an effect on the Day 1 battle. It was the rifle that was issued to Custer's troops, the Carbine was not manufactured before July 1863.
The Lincoln story is based around the second visit by Spencer to the President. On the first visit, the rifle had mis-fed and Lincoln decided that it was not appropriate for troop use. In August of 63, Spencer returned to the President for a second demonstration which changed Lincoln's perceptions and opened the door for wider military issue. The rifle used at that second demonstration is still at the Smithsonian, and was included in a display of Lincoln artifacts until the opening of the new Americans at War display. I am not sure where that rifle is now, and do not believe it is currently on display.
TomH