I own a number of binoculars, most in the middle to upper end. I have four Zeiss bins, the 10x25 Victory, the 8x32 Conquest, the 8x42 HT's, and a pair of 8x42 Terras. I also own the excellent Canon 10x30 IS bins, Nikon 8x32 SE's, and Vortex 6x32's.
I would suggest looking at the Zeiss Terras. They are particularly nice at your price point. So are Nikon Monarch 5's.
FWIW, the Terras are what I keep in my vehicle as part of the always have a pair with you. However, when hiking, it's almost always the excellent Zeiss 10x25 Victorys but they're well above your price point.
8x32 or 8x42 are probably the best all around size, with a nod to the 42mm objectives for darker times of day and, especially, astronomy. Probably stay away from 10x unless they're stabilized or you can hold them very steady. Jittery images sort of ruin the experience.
BTW, I wouldn't worry about correct vernacular. Most everyone refers to them as bins, binoculars, a pair of binoculars, etc.