The paper also discusses the EL1:4 boundary that I use to mark the start of deep space, which lies at 152066 km from the Earth's center.
To be strict, though, there are several senses of the word orbit. In its broadest sense, an orbit is any trajectory of an object in the local gravitational field. If you stand on the Earth and throw a ball in the air (or make an apple fall from a tree) its path (neglecting aerodynamic effects) is not, as you were taught in school, a parabola. It is in fact a very eccentric ellipse with the perigee extremely close to the center of the Earth. The ball will complete a very small part of this orbit before the surface of the Earth gets in its way. In this sense, `suborbital' rockets are in fact just orbital rockets with negative perigee height.
Because even many people in the space field don't appreciate this, it's worth doing the worked example. Suppose we throw a ball at a 45 degree angle and a moderate speed of 50 km/hr (13.9 m/s); let us assume we are in Harvard Square (42.4 deg N) and throw the ball due eastwards. The resulting orbit is -6371.8 x -10.0 km x 42.4 deg, an eccentricty of 0.998. (As noted in the section on apogees and perigees, we quote the heights relative to a fictitious 6378 km spherical Earth, and due to oblateness Harvard Square is 10 km below the zero surface.)
At the other extreme, an orbit around the Earth can also be hyperbolic, with eccentricity greater than 1. When a rocket reaches escape velocity, many writers say `it is now in solar orbit'. From the point of view of calculating its trajectory in an N-body solar system simulator that's not the right way to think about it. The object is travelling away from Earth very fast, but it's still deep in the terrestrial gravity well, far below the geostationary satellites, and the differential effects of the Sun's gravity are negligible. Only several days later does the object reach the edge of the Sun-Earth Hill sphere where it is a better approximation to model it as orbiting the Sun. Until then, I consider it to be in Earth orbit, albeit an unbound one on an escape trajectory.
Different categories of orbit (LEO, GEO etc) are noted in the catalogs and defined in the Orbital Categories section.