JD- there are several stocks for 1903 and 1903A3 rifles
Very roughly, there are four main types:
"straight" stock (sounds like the type you have)
"Finger groove" stock (such as your friend showed you)
"C" stock (has a pistol grip, was used on sniper models [1903A4] as well)
"Scant" stock- attempt to make a 'C' type stock from 'straight' stock blanks, has a slight pistol grip
Changing the stock may- or may not- harm value. What you need to consider is whether or not the stock you wish to use is USGI issue, or a commercial replacement, and whether or not the stock is appropriate for a 1903A3 or not.
The symbol that is confusing you, that you see on the Underwood carbine and the 1903A3 is the "flaming bomb" Ordnance symbol. It is not a maker's mark, it is a US Ordnance stamp. It is an extremely common marking on US martial arms of the era you are involved with on these rifles. In broad general terms it means the object had been accepted or approved for use in the military by the Ordnance Department. Other marks are preset that mean roughly the same thing on various rifles, etc.
For your stock, I don't know how to determine what the first and original stock type was on the rifle. Most likely it was a straight type or scant type. Perhaps somebody knows s/n ranges that had one type or the other but I do not.
One thing that can help determine authenticity or originality are the 'cartouches' that are on the stock. These are stampings applied by the US military to indicate that the rifle had been inspected and accepted. Look closely at the stock you have, and see if you can spot any initials stamped (not carved!) into the stock, perhaps surrounded by a stamped rectangle, and also look out for a 'cossed cannons' cartouche, which will seem to be a symbol like a capital O with an X over it, with the legs of the X extending past the edges of the O.
Here is an example of those types of cartouches on an M1 rifle stock. 1903A3 ones are similar (not my rifle, but I wish it was)
Also, there were 'rebuild' cartouches, denoting where a rifle had been rebuilt by the military. Usually, the original cartouches were sanded off before the rebuild one were made. On my M1 carbine, an example is the 'AA' stamp shown here:
There are several places to get 1903A3 stocks. This ad claims that they have unissued C stocks. This will mean they bear no cartouches:
http://northridgeinc.com/store/index.cfm?p=c156/i416
Likewise here are straight stocks from the same folks:
http://northridgeinc.com/store/index.cfm?p=c156/i416
lastly scant stocks, same people:
http://northridgeinc.com/store/index.cfm?p=c156/i416
These folks used to have great 1903 and 1903A3 stocks but they are all gone. However, they have misc. parts, and I feel they are good
www.thecollectorsource.com/page/1219295
The 1903A3 is the first rifle I ever fired - at age nine. My Dad still owns it, and it's a great rifle.