Thanks to a very helpful suggestion in the comment, I was able to do an initial test with an online orbital simulator. It can be found here:
This starts off (helpfully) with the Earth moon parameters, for the mass ratio and the spacing. All we need to do is tell it where to put the initial particle. Here is the screen where you can do that:
This is "details of initial situation". As a side note, I believe they marked both the system CM and Earth, but they partially obscure each other.
You can see that I've changed the parameters to what we need for simulation starting at L1. It is around 84% of the way from Earth to the moon. Make sure the angle is set correctly too. Then we have two parameters for initial velocity - set both to zero because we are in the co-rotating reference frame.
Here's what I get doing the simulation:
You can see that it initially moves tangentially to the Earth-moon line, and then crashes into Earth.
I can't tell if this is the answer I'm looking for. Simplistically, I don't see how it can move in that direction. Given the shape of the saddle point, I believe it should move either to the right or the left in the above plot. But the mechanics in the co-rotating frame are notoriously spooky. It still seems possible that this is right, but I can't tell.
The problem was that I misunderstood what the input parameters were. The initial tangential velocity is still given in reference to a non-rotating frame. In order to be "stationary" relative to the Earth-moon system, this parameter needs to be equal to the radius parameter. See help:
with respect to the Centre of Mass, given in units of the oribital speed of the Moon
ditto, positive velocities point in the same sense as the lunar motion
So in the above L1 example, it the initial tangential velocity needs to be 0.84. If you do that, you can get this:
This is much more reasonable, and more likely correct. In this example, it does crash into the moon, but only after about a month. The scenarios appear to fit:
- L1 moon-side drop: orbits for 1 month, crashes into moon
- L1 earth-side drop: goes into high Earth orbit, looks stable
- L2 moon side drop: goes into moon orbit, can transition into high Earth orbit
- L2 far side drop: orbits both Earth and moon at very far distance