If I want to spend $150-250 on a keyboard, should I get a MIDI keyboard (like something from M-Audio) or an electronic keyboard type of device (like something from Yamaha)? Can most electronic keyboards be used as MIDI controllers?
It is rare to find a new electronic keyboard that doesn't also have MIDI. So if you want some basic sounds and the ability to play away from your computer, then go ahead and get a keyboard with sounds built into it.
Dedicated MIDI controller keyboards may offer more control via knobs, faders and sometimes pads. If you need any of that stuff, then you may prefer to go with a MIDI controller.
The third large category that you are missing is used keyboards. Check your local source of used gear (ebay and craigslist are my usual starting points), and you may find some excellent values.
Personally, I find that the midrange and high-end synths of a decade or two ago have vastly better feel than comparable modern controller keyboards. My Yamaha EX-5 has 76 keys and a better keybed that any controller keyboard I've used. Similarly, the hammer-action Roland and Yamaha keyboards of the past are far better than any of the 88 key controllers available new now.