Glad you got it to work! Going back to your original attempts, here’s what may have been the source of your struggles:
In SketchUp, circles are actually represented as many-sided polygons. When you drew the starting circle, it is almost certain that none of these sides was oriented in the vertical, blue axis. For example, if you start the circle on the red axis, SketchUp will put the first vertex on red axis and draw the first edge as a chord to the true circle - which puts it at a slight angle toward the blue axis. Then when you pushpull the circle to a cylinder, the side of the cylinder is actually a many-faceted surface, one face for each side of the original circle. You can see the facets if you turn on View->Hidden Geometry. And, like the edges, none of the facets are parallel to the blue axis.
Now, when you activate the rotate tool, it tries to lock onto one of the facets unless you use one of the techniques provided above. That will put the rotation off axis because the face is off axis. In the animation below you will see that the rotation cursor is black, which indicates it is not aligned with any axis. Rotation about that skewed direction will cause what you described.
