- the planet's orbit being eccentric, not a perfect circle
Apparently @DavidHammen knows just enough about things like this to be dangerous. I can't speak to the topic mathematically, but in the figure the little blue cigar is the Earth oscillating towards and away from the Sun as it moves in its slightly elliptical orbit, and the cool looking orbit is in this case SOHO, who's been in a halo orbit for more than 20 years, faithfully executing it's station keeping maneuvers towards or away from the Sun, except when it didn't (see Roberts 2002 linked there).
The near-rectilinar halo orbit of the proposed, future
stargate err... , wormhole, err... gateway will be a (near-rectilinear) halo orbit (1, 2 and answers therein) around the Earth-Moon L1 and/or L2, and the eccentricity of that is much larger than that of Earth's heliocentric orbit ($\epsilon$=0.055 vs 0.017).
I always get my science fictions mixed up. (Space Force!, Rocket Racing!, etc.)
Also, Chang'e-4's radio link to the Earth Queqiao relay satellite will be a the Earth-Moon L2.
So for significantly elliptical systems, yes. For very significantly elliptical systems, someone else will have to answer about the ER3BP, and of course, someone has already asked Are there any natural circular orbits? to begin with.
- The two massive bodies being of comparable mass, orbiting a common barycenter (though both in circular orbits). Specifically, how would they morph/move?
In this answer I show just how to calculate the positions of L1 and L2 for two masses. I'll try to make a plot of how the two points move as the ratio of the two masses changes, that will take a half-day, need to get a stable 10-20 before looking at the stable points.
GIF: SOHO orbit, data from Horizons, plot from here
below left: Top-down view, eleven years. right: View from the side, one year. (Sun to the left) James Webb Telescope prototype orbit, data from Horizons, plot from here