Number of intersects two circles can have

GRE Quant Problem Solving
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Maximum number of intersects two circles with different centers?

I think it should be 3, but according to Quiz 11# (Geometry column 2) question number 24, it says max intersections can be 2.

please clarify my doubt, above I have provided a sketch for my justification showing 3 intersection points.

You might counter by saying the middle intersection in the figure is not an intersection but just a tangent / circles are just touching, but similar situation occurs in quadratic equation when discriminant is zero and the parabola has only one root, we still say parabola has only one intersection even though the parabola is tangent to x axis.

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But such a circle can’t exist. If two circles match at three points, they must match everywhere.

I think your image is just geometrically misleading. How about using this image as a reference instead:

image
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You can’t enlargen the inner circle such that it matches at two more points.

3

Also, in your diagram, the bottom line is incorrect.
Just because they are thick lines, it looks like it is intersecting at the bottom too.
However, in reality it doesn’t intersect at the bottom.
Only the left and the right points.

There is a vertical distance between the centers at the bottom.
Try using Desmos, or Quemath to plot 2 circles and see.