Exactly! It's n+1 points in n dimensions (when finite). Another way to think about it (the way that I know because it extends into general Banach spaces and not just n dimensional spaces) is that each point inside is the unique weighted average of the extreme points (corners). So in 2d, if you have a square you can get that middle point by averaging all the corners, or averaging two opposing corners, so it's not a simplex.
Understanding that 10 years ago might have made it easier for me to understand collision detection algorithms for game physics