Charlie - a good example of why they say a picture is worth 1,000 words. I can picture what you are describing, in general terms though. Your first description sounded, to me, like a good description of a metal-cap bullet, which was used in .32 auto. I guess I leaped upon the words concerning the exposed core, and thought you were referring to it being exposed above the case mouth, which cores of Metal Cap bullets often are, as opposed to full-metal jacket bullets, which are not. It did not sound like a “normal” bullet - that is, a FMJ bullet with the jacket going down into the case to the very base of the bullet. From there, the treatment of the base, or bottom, of the bullet varies greatly from factory to factory. On some bullets, the jack goes to the bottom of the base but does not fold over the edges of the core. On others, it does fold over the edges. Some had flat bases and some have concave bases.
It sounds to me like you have pulled the bullet out of one of the rounds, at least. Your first description, again to me, sounded like you were describing the bullet still completely seated in the case making a normal OAL loaded cartridge.
If it as I think I gather now, that the jacket goes clear down to the bottom of the bullet (the base), then I apologize to Vince for challenging his assertion that you were describing a normal .32 auto bullet, although I still hold to the fact that there are many, many different “standard/normal” bullets for the .32 auto, just as there are for many calibers of ammunition.
To answer your question adequately would require pictures of what you have, and maybe require the pulling of bullets out of dozens of specimens to see who did what. Someone else would have to do that.
Sorry that, as it turns out, it seems I was of little help. The .32 is a favorite cartridge of mine, and because I have a very incomplete collection of somewhat over 1,000, to answer your question technically and correcting would likely be a daunting task. I wish I could do it because I think it would be very interesting, but I have neither the time nor the number of different duplicate rounds to make a meaningful study out of pulling them apart, which of course, I don’t care to do with my collection rounds. Not that I never had, but not in the quantities it would require.