For ID purposes only, you'd probably be fine with most superzoom or even travel zoom P&S cameras...though if you're shooting from inside to a feeder, and might encounter lower light conditions, you might do better with a slightly larger sensor and a little less zoom.
Most superzoom P&S cameras, with lenses that reach equivalents of 1,200mm to 2,000mm, use tiny tiny sensors which work fine in good sunlight, but will struggle to get details in lower light or even overcast...so it could hinder identifications if your light isn't always going to be bright...and being 20 feet away, that's a lot more zoom than you really need. You'll probably want about 300mm to 400mm equivalent reach to really fill the frame with a bird 20 feet away - and some cameras to consider would be those P&S cameras that have 1" sensors...probably the only one that gets close to your price range would be the Panasonic FZ1000...that will give you a decent 1" sensor plus up to 400mm optical equivalent reach...that should handle pretty much any identification needs in most lighting conditions.
If you think the light is going to be brighter or more reliable, then consider any of the other superzoom type cameras with smaller sensors such as the Panasonic FZ80, Sony HX350, Canon SX540, Nikon Coolpix B700, etc.