Those guns were all marked simply "Colt D.A. .38" even though the later ones would take (and were intended to take) .38 Special.
The Army contract Model of 1903 was chambered for the .38 Special and the civilian models from 225xxx ("Navy") and 222xxx ("Army") were also made for .38 Special; that was from about mid-1904. So since production of that model series finally ended in 1910, production of .38 Special guns lasted around 5 1/2 years.
Colt was not noted for customer oriented caliber markings. Their .32 Pocket revolver was made in both .32 Colt and .32 S&W/.32 Colt New Police, but the marking is just "Colt D.A. .32" with nothing on the gun to indicate which .32. The New Service .45 was marked "Colt D.A. .45" for .45 [Long] Colt but the same marking was used for the Model 1917, which was made for .45 ACP.
Another example is the Colt "Marine Corps" Model. A 1905 Colt ad says specifically that it is for the .38 Long Colt AND the "38 S.& W. Special". A 1906 ad for the same model says it will "take the following cartriges: .38 Long and Short Colt, .38 U.S. Government, and .38 S. & W. Special." Yet, all those were marked "Colt D.A. .38."