In addition to Rashmi’s excellent and detailed response, the point about the pictures is that (exactly as you say) everything here really involves hundreds, thousands or even millions or billions of dimensions. Unfortunately, you can’t draw pictures in more than 3 dimensions and our poor human brains (or at least my poor human brain anyway 
) are only capable of visualizing things in 3 dimensions. So the pictures are a very very limited attempt to give some intuition using the extreme limitations of visualizing things in 3D. In addition to all the great links that Rashmi included, here’s a thread that has some more discussion about local minima and includes a link to a paper from Yann LeCun’s group about the complexity of loss surfaces.