Note: for lots of great background related to this question, see lagrangian points - The design of the halo orbit of the James Webb Space Telescope - Space Exploration Stack Exchange
There are numerous references saying that JWST will be in a halo orbit around L2, meaning that the period of the oscillation in the Y and Z axes of the Rotating Libration Point (RLP) coordinate system are equal. Thus, as viewed from the Earth, the orbit would describe a similar oval over time.
But according to Seasonal variations of the JWST Orbital Dynamics, launches of JWST before 13:00 UTC (Update - note this is for October launches) tend to lead to non-halo Lissajous orbits. For them, the oscillations in Y and Z have different periods, and the orbit looks from Earth more like a series of ovals offset in time, like this:
(But note that JWST orbits L2, not L1 like DSCOVR's orbit pictured above.)
Page 3 of Solar Radiation Pressure Effects on the Orbital Motion at SEL2 for the James Webb Space Telescope has a beautiful display of the various orbits based on modeling of various launch times on 2021-01-14. (Update: note that the 12:20 launchs in mid-January are clean halos)
The actual launch was 12:20 UTC on 2021-12-25,
implying that it would not be a halo orbit.
So what is the expected orbit, and what are the implications for station-keeping, propellant and mission length, etc? E.g. how does the JWST avoid the dangerous region close to L2 where it would be in the Earth's umbra and encounter power and thermal challenges?
Update I see that I repeated a quote out-of-context, and it is now more clear to me that the resulting orbits vary by both month as well as time of day. I've added some updates related to that above. Based on the January 14 plots in the 2nd paper on Solar Radition effects, I'm guessing that the Dec 25 12:20 launch is actually a pretty clean halo orbit. But I'm still interested in confirmation of that and other up-to-date details on the predicted orbit.