How can orbiting be explained in terms of falling, projectile motion, or circular motion?

Well, orbiting can be explained in generalized projectile motion, because orbit is a motion of an object which is subjected to gravitational forces alone.

Notice I used the word generalized?  Our parabolic projectile motion equations only apply in uniform gravitational fields.

At the level of orbit, gravitational fields are spherically symmetric, but not uniform.  For this reason, we generalize the projectile motion formulas to apply for gravity varying by Newton's law of gravitation, near a spherical massive body.

In a special case of orbit, the object is traveling perpendicular to the line joining it and the nearby massive body.  And it is traveling at the correct speed, such that the centripetal acceleration equals the local gravitational field.  In such a case, the orbit is perfect circular motion, constant speed, circular path.

In the general case of orbit, the path is an ellipse, with the massive body at one of the foci.  If the bodies are of comparable mass (such as in a dual star system), it is the common center of mass at one of the foci of their individual elliptical orbits.  Such an orbit is governed by Kepler's laws, which can be derived from Newton's laws of motion and gravitation.