Types of cameras and manufacturers opinions are going to be as varied as the inspectors in this industry...
With that being said, I personally look for a camera with super zoom capability, long reach flash, and uses common AA rechargeable batteries. I have enough proprietary chargers and equipment cluttering my desk, so any chance I can find equipment that uses standard batteries then I am interested in making a purchase.
I am currently using an Olympus model SP-620UZ with 21x optical zoom and see Mars surface digital zoom capabilities
It captures fantastic photos and has amazing battery life using 4 AA sized batteries. I typically capture between 300-500 or more (yeah I'm that guy) photos per inspection and batteries will last at least 2 days of inspections between charges even when I must use flash.
My back up camera is a Sony DSC-H2 which they don't make anymore. It is a very nice camera but is a battery hog using 2 AA's which I would have to change at least once during an inspection. Most of the Sony DSC-Hxx cameras now use proprietary batteries, but I still love Sony technology and quality. You can still find a few used or barely used of the H2's and H5's on Amazon and Ebay.
You can typically find decent quality super zoom cameras new between the $200-400 (sometimes less) range. Black Friday deals will be abounding with some great deals. The Olympus SP620UZ is in the lower $200 range or even less if I remember right.