I wish I had one of those for the last illuminated keyboard I had until finding out it had a hidden defect when first bought new until it gave out one day. That one would pull quite a bit on a port like I suspect could very well be happening with the optical drive there.
The other thing mentioned as well is that the ports simply may not supply enough for some reason like the other laptop does. That would easily explain why it works on one and not the other. Plus the one you have there is a Netbook according the page at Dell found for it. Inspiron Mini 10 (1012) Netbook - Long Battery Life & Colorful | Dell
A regular fuller sized laptop would supply adequate power since people plug printers and other usb devices in on a regular basis while netbooks as a rule are the newer kid on the block so to speak and are meant mainly to be a carry everwhere convenience rather then a workstation at home, work, or school. Kind of puts you in a bind in that sense.