Hi, Irishbeener,
Takedown of the Model 1903 .32 ACP does not involve the button below the barrel. To take the pistol down, cock it and remove the magazine. Then retract the slide to the point where the arrow line on the right front of the slide lines up with the frame, then turn the barrel counterclockwise (as looked at from the front of the gun.) With the barrel turned, the slide should come off the frame. The barrel can then be turned back and removed from the front of the slide.
That should be enough for normal cleaning. Further disassembly is NOT repeat NOT recommended; that gun comes apart easy and goes together hard.
(The trick of pushing in the front "button" applies to the dual link Colts, like the other 1903, the .38 ACP hammer gun.)
Two comments: First, the .32 ACP is not actually rimless; it is semi-rimmed. Second, the early .32 and .380 frames are not identical except for caliber; the mag well of the early .32's is smaller and too small for the .380 magazine. Later, Colt apparently made the frames identical.
Also note that the model numbers (dates) have been assigned by collectors; Colt never used them.
Jim