Long-time lurker here. I own 2002 Prius. I'm trying to replace my auxiliary battery. The battery shown on the left in the attached image is the one I'd been using. The one on the right is the battery suggested by Toyota on the basis of my VIN, but two mobile mechanics told me it's the wrong battery. I'm really confused. Thanks for your help.
The Toyota battery on the right is correct. 2002 Toyota Prius Vehicle Battery - 2880021050 - Genuine Toyota Part Any 12v AGM car battery with the right size terminals will do the job, the trick is to find one that will fit in the small compartment in the trunk. The battery I use isn't OEM, so I had to cut away part of the battery hold down bracket to make it work.
I noticed in my blue car it had the bigger battery, which required a bigger hold down bracket to fit, vs the smaller one in all my other cars. Also, to get the bigger one in/out you have to remove the plastic tube duct thingy that runs above the battery, otherwise it would NOT pull out of that spot. A mazda miata battery will fit and has the vent hose to vent outside the car.
I have no idea how the battery you've been using fit in that tiny little space. Not very well is, I suspect, the answer. Nor would I guess how the cables reached unless they were modified. Yes, the narrower battery on the right is correct.
See this thread on the larger size battery and the toyota TSB that recommends a bigger one be installed. Aux battery specs NHW11 post-TSB | PriusChat