That was it!! I took a peek at the armored_vest mod and noticed that the spritesheets were 512 (16 pieces) wide. After changing to 16-wide sheets, it works perfectly!
In retrospect, it might have worked if I had listed my original image dimensions correctly in the extraSprites ruleset.
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It could be something with the spritesheet or drawing routine. I copied the code directly from the original armors and made only the relevant changes. There is nothing listed for drawing routine in the original ruleset, which I assume is because it defaults to 0 which is the correct one for soldiers. I haven't set a drawing routine either. Also, I think it is using the correct drawing routine because it has more than half of the soldier drawn correctly and no parts drawn in the wrong place.
I'm using a 256x1400 and 256x1440 spritesheet (8x35 and 8x36 pieces) which is what Falko's tool generated for me when I told it to combine the images given to me by XCOM Extractor. Is that the wrong dimensions? And if so, what are the correct dimensions?