Camera calibration using checkerboard instead of 3D calibration object
Now with a checkboard given one image, I obtain the internal camera matrix and external kamera matrix.

No, you don't. You get a single coordinate transformation matrix that contains only the ratio between focal length and object distance. You cannot split that into an internal camera matrix (focal length) and external matrix (object distance) without further information.
If you want that extra information from only a single image, then the points of your calibration target must not be all in a single plane. This is why people use cuboid calibration targets. Alternatively, you can take at least two images of checkerboards from different perspectives, which also allows you split focal length and object distance.

Accordingly, am I right that it is necessary to do both, the calibration with the checkerboard to get an accurate estimate of K and then the calibration with the 3D calibration object to obtain the extrinsic camera matrix, so that the 3D calibration object defines the world coordinate system?

No, this is not necessary. A single image of a cuboid calibration target is enough to determine both the internal and external camera matrix and a single image of a checkerboard gets you nothing.