My understanding is that geodesic is measured on the surface of the ellipsoid and is the always the shortest distance on the earth. Planar is just straight line distance. Is this the correct interpretation?
Yes, broadly speaking. A geodesic (in mathematics) is the line of shortest distance on a given manifold. In geography our "manifolds" are usually either 2d (or 3d) flat space as an approximation to the shape of the earth for small distances, a perfect sphere, or an ellipsoid. Any shortest lines in these spaces are technically geodesics.
So on a flat space the geodesic distance is given by Pythagoras' theorem (which generalises to three and more Cartesian dimensions), but it is rarely referred to as a geodesic distance. On a perfect sphere the geodesic lines are parts of "great circles". On an ellipsoid...its complicated. There are also geodesics in higher dimensions and on surfaces with negative curvature (hyperboloids), and geodesics in 4d space-time but then you get into Einstein and Relativity...
Just make sure whatever tool you are using is clear about what distance measurement it is computing. Some tools may automatically switch to great circle or ellipsoidal geodesic if your coordinates are in lat-long degrees, otherwise it will use Pythagoras. Other tools may secretly project non lat-long coordinates to lat-long and use great circles, or not.