OK, I misinterpreted your original screen shots which seem to actually be Acrobat showing the two page spread, but you've cropped it to the point where you can only see the bleed area. In the top shot with the white line, the image on the right hand page extends partly into the the left page in your layout, I would surmise, but not as far as the bleed allowance. I'm surmising the image of the elephant is a true cross-over that extends beyond the bleed allowance. Is that correct? If so, then what you are seeing is, indeed correct. If you add crop marks to the PDF you'll see the duplications should be trimmed away.
The directions in your post above regarding the bleeds for perfect bound are horrendous, and virtually impossible to work with for crossovers. Essentially they are telling you to make your pages .125"/3mm wider and have no inside bleed, but to leave a blank gutter at the spine. To properly prep those spreads you will need to place the image once the way you want it relative to the outside margin of the page that holds the larger piece, then split the spread (see Salah's link).
Now copy the image on the half of the split spread where it appears and paste in place on the other half, so you have two copies of the same linked image. Shift that second copy away from the gutter twice the margin width (.25" or 6mm). Now crop each copy back to the inside "page edge" which is really .125/3mm inside the page (I'd either set the margins for this, or more likely add a guide on the master pages) so it does not extend into the glue area. When you export with the inside bleed should be set to 0 and the glue area should now be blank.
It would actually be much easier if you used an inside bleed. Just split the spread and duplicate the cross-overs as above, but there is no need to shift the second copy. Crop both back to the real page edge, leaving the inside bleed allowance blank.
