That qualifier is indeed crucial.

A more technically correct term would be "spacelike line". (It's a line and not a 3-surface because it is the locus with r = 0.) But I think "moment of time" is easier to understand for an audience that is likely not to be familiar with GR technical terms.

> different infallers can arrive at the singularity at different times

If we're going to that level of detail, then there is no invariant way to specify "at the same time" inside the horizon anyway, since the spacetime is not stationary there so there is no timelike Killing vector field and no set of spacelike surfaces picked out by the spacetime geometry as surfaces of "constant time". So "at different times" and "at the same time" inside the horizon are just a matter of coordinate choice and have no invariant physical meaning either way. But again, that's probably too much detail for people who are not familiar with GR technicalities.

> For a BH that forms by gravitational collapse and eventually completely evaporates, surely the singularity has a timelike worldline?

Not for the case where the final black hole is a Schwarzschild hole, no. The locus r = 0 is a timelike line at the center of the collapsing matter until the collapsing matter's surface reaches r = 0; up to that point r = 0 is not a singularity and is a perfectly ordinary location inside a blob of collapsing matter. But once the collapsing matter's surface reaches r = 0 and forms the singularity, the locus r = 0 becomes a spacelike line, and that's the spacelike line that anything falling in after the collapse has finished will end up hitting.

> Maybe easiest to think about that if the singularity is not always at the spatial origin of a system of coordinates

It's not at the spatial origin of any system of coordinates. That's impossible for a spacelike line.

> if SN1987a's remnant contains a black hole, that black hole is surely moving around the galaxy with the luminous matter "now"

It depends on what you claim is "moving". You can certainly pick out a "world tube" in the external spacetime (say by taking a timelike 3-surface formed by the dynamics of a surface of constant r just outside the horizon) and say that "whatever is inside the tube" is moving around the galaxy, because that's what the world tube describes. But you can't say that the singularity is "a point" that is "somewhere" inside that world tube. The singularity doesn't have a spatial location, and spacetime inside the hole's horizon simply doesn't work the way the ordinary language that we use to describe objects moving in space through time assumes that things work.

> the usual Carter-Penrose BH diagrams showing long horizontal wavy line segments for the singularity are misleading because of exaggerations caused by the particular conformal chart it uses

They are not misleading about the fact that the singularity is spacelike. And if you look at a Penrose diagram of a model like the 1939 Oppenheimer-Snyder model of collapse of a spherically symmetric star to a black hole, you will see that the line marking r = 0 has a "corner" at the upper left of the diagram where it changes from timelike to spacelike; that "corner" is where the singularity begins.