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I think when the png is being constructed it doesn't handle the interaction between the opacity of the implicit plot and the point. Try adjusting your opacity. I also tested this:

distance = 56; x,y,z = var('x y z')
mypic = implicit_plot3d(x^2 + y^2 + z^2 == distance, (x, -10, 10), (y, -10, 10), (z, -10, 10), opacity = 0.5)
for k in range(-9, 10, 2): mypic = mypic + point3d(vector([5, k, 4]), color = 'red', size = 10)
mypic.save('mypic.png', compress = false)
mypic

The inline image shows a line of six red points from up-left to down-right ( \ ), with a center gap where four are completely blocked by the sphere. The saved png shows all ten points in a line from down-left to up-right ( / ), with the center four slightly hazy. So the png is also swapping x with y for the points (the axis labels are correct).