They're not lens flares, but they're still an artifact of the system, which is probably what most people mean by "lens flare" anyway, due to lack of a better common term.
According to the Wikipedia definition, I'd say that Webb's artifacts could be classified as "lens flares".
>This happens through light scattered by the imaging mechanism itself, for example through internal reflection and forward scatter from material imperfections in the lens. Lenses with large numbers of elements such as zooms tend to have more lens flare, as they contain a relatively large number of interfaces at which internal scattering may occur.
Diffraction spikes are caused before the lens gets involved; improving the lenses cannot remove the spikes (though there may be structures causing diffraction spikes in-between lenses).
The spikes from JWST are primarily caused by the edges of the mirrors and the three support struts.
There likely is some amount of lens flare (though I don't know if it is significant, the optics are Very Good), but the dominant artefact is the diffraction spikes.
