Halo orbits are a sub-class of Lissajous orbits.
So that image showing a simple circular-ish orbit is just showing a 1:1 Lissajous pattern.
These Lagrange-point orbits are really orbiting around the larger body, in a way that's resonant with the smaller body. If we talk about the Earth-Sun system, then satellites like DSCOVR, SOHO (L1) and the (hopefully) upcoming James Webb Space Telescope (L2) will be in Heliocentric orbits (orbits around the Sun) about 1% closer or farther (respectively) than the Earth's orbit around the Sun.
The Earth's gravity is weak there, but strong enough to "tug" the satellites along a little faster or slower to keep them synchronized.
When you step into the twilight zone of a rotating frame and move with the Earth, their motion appears to be around the L1 and L2 points from your point of view in the rotating frame.
Mathematically, when doing calculations for a simplified Circular restricted three body problem(CRTBP, CR3BP) the equations become easier when you use the rotating frame.
In an inertial frame, those satellites will appear to drift slightly up and down, making one full cycle roughly twice a year. In the rotating frame only, that motion looks like an orbit around, or at least associated with, the Lagrange point.
That motion has a "horizontal" or left-right component, and a "vertical" or up-down component.
In some cases, when this CR3BP motion is large enough amplitude, those motions can have the same period, and so the orbit will appear to be closed, cyclic, and periodic in the rotating frame. Orbits in this subset are called Halo Oribits. SOHO and the future JWST will be in these.
However there are plenty of orbits in this family where the horizontal and vertical motion do not have the same period, and so they will appear to make a criss-cross or Lissajous figure in space viewed in the rotating frame. These are called Lissajous orbits. There is no special relationship between the horizontal and vertical periods, they don't need to be locked in say a 4:3 ratio for example. Remember these are not real orbits.
From a satellite's point of view Halo orbits are used because they tend to circle around the Sun-Earth axis (or the Earth-Moon axis) and avoid radio interference and power outages due to eclipsing of the solar panels. DSCOVR's orbit will put it in it's Sun Exclusion Zone in about 2020 where the communications line of sight will be too close to the Sun, so there is a planned orbital correction there to handle the situation. You can see from the image the insertion point labeled LOI, and about a dozen cycles in five years. The horizontal and vertical periods are almost the same for this orbit. From Lissajous Orbit Control for the Deep Space Climate Observatory Sun-Earth L1 Libration Point Mission
After 2020, DSCOVR will have to burn fuel every 3 or 6 months to stay on that ellipse and avoid the Sun exclusion zone, which will then run out around 2028.
To read further about equations for calculating Halo orbits and some Python, see
Also see these questions and their answers: