Yamaha and Tascam make portable recorders in that price range ($100-$150). Most portable recorders record in a stereo XY pattern,

However if you are trying to record sound effects with something like that I don't think it's very feasible. You have to be able to record those effects very cleanly. If you can hear the room, ambient sounds, or other stray noises they're going to become very prevalent very quickly in a game as the effects usually play many times during the course of a game. There's also all the editing you'll have to do in terms of EQ, levels, and noise reduction. This is one of those ideas that sounds simple but is actually very complicated. You'll be better off hiring someone unless you're ready to spend a lot of time and more money learning to record foley.

going to staunchy and respectfully disagree with that :) as i honour no profession nor protect their interests..

electret mics are extremely inexpensive and have excellent "detail" - it's not the price of the gear, it's what you know about it... "realistic"/radio shack used to make stereo electrets for a few $ - if you check to make sure there's no ground loop/buzz (easy with battery power) you can create excellent foley with them.. mics have different characteristics and may take some post to get them to sit (low pass for less brightness/presence, a bit of compression to limit dynamic range) as one "scene".

for mono i use an oktava mk12 for general stuff.. classic, probably under $100 nowadays. but shures will sound about the same really. minor details. as long as the mic doesn't hum..

experience is certainly an asset, but i encourage a "can do" attitude..... at the end of the 90s chinese factories (eg. for behringer) started building copies of expensive mics and the market got shook up.. nowadays, you don't have to spend a lot for clear audio... but you still have to pay attention.........

eg. a pop screen for plosives or similar really is trivial to construct with any porous material (i use shadecloth, big holes, still stops plosives).

for mono i use an oktava mk12 for general stuff.

An Oktava MK12 is about $200 used, a matched pair would be more like $500-$600, it's a nice mic but not the best for mic for foley especially for the price.

What would also be the correct term to search (english is not my native language)

Portable Digital Audio Recorder is correct.

Here's an example of something in your price range

http://www.sweetwater.com/store/detail/DR22WL