Great question!!
"Koopas Ly"
During a crosswind landing, for instance a left crosswind, you'd lower
the left, upwind wing to counter the right drift induced by the
crosswind.

You'd also use some right rudder to keep the nose straight and prevent
it from "weathervaning".

This is a proper xwind landing conditon and is a slip (forward vs side is
debatable). You are banking left into the xwind to counteract the drift and
get the a/c to track down the centerline. You apply enough righ rudder to
align the nose with your direction of movement (i.e. the centerline). I
would not call this weathervaning.

Is this "weathervaning effect" caused by your leftward relative motion
due to the left bank OR by the rightward crosswind ITSELF?
Personally, I think that the former applies. The rightward crosswind
only displaces the airplane to the right. Only the relative motion of
the airplane with respect to that airmass would induce the
weathervaning effect. I presume that the airplane does not know,
aerodynamically, of the left crosswind.

I'll leave weathervaning undefined but the last line is correct - as long as
the a/c is airborne, it does not know of the left crosswind. The xwind is
relative to the ground and if you are not touching the ground, then you
might as well be at 30,000feet

Next thing I was wondering, which is related to the above: say you're
dead on centerline on landing, and all of a sudden a crosswind from
the left starts blowing. The effect would be that you should only be
displaced to the right of runway centerline. Your airplane nose would
still be parallel to the centerline. Do you agree?

Yes & No. The the a/c will weathervane into wind. That is, the nose will
tend to point into the wind. The wheels represent the vertical axle of the
weather vane. Assume 1 wheel (glider) on the CG. You could grab a wingtip
and rotate with little effort. The wind hits the side of the aircraft. End
of of the a/c with the most side area will be rotated downwind. The tail,
like feathers on an arrow, will always be that end.

Now the whole a/c will also tend to be displaced left to right. But imagine
same single wheel and try to grab a wingtip and drag the a/c against the
friction of the wheel. You won't be able to budge it.

So, the net effect is that the weathervane effect will be immediate and
dominant with weight on the wheel and low speed. At higher speeds and
lighter weights, the drift will take over. What happened to weather vaning?
It is still their but the forward motion of the a/c changes the net wind
vector and the a/c ends up 'weathervaning' closer to straight ahead than
towards the xwind.

A gllider tow from a dead stop in a xwind demonstrates all that very nicely.
As soon as you start rolling, balanced on one wheel, the glider will rotate
(vane) right into the wind. So downwind rudder is required at first. As
speed picks up, both the rudder becomes more effective and the relative wind
vector shifts towards your intended direction of flight so less downwind
rudder is required. As the wing starts lifting the weight off the wheel,
the glider starts drifting downwind with the tire scrubbing sideways.
Dipping the upwind wing fixes that and you leave the ground in a slip. As
soon as the wheel is off the ground, you can basically let go of the
controls for a second and the plane will assume an immediate wings level
crab relative to the ground and off you go. This is less complicated than
it sounds but you have to do it a few times before you can do it right.
Until that time, the pull of the towrope on a quickly accelerating tow plane
will keep you out of the weeds while you regain your dignity. (BTW, all that
happens in reverse on landing but again, the stars are favorable and
decreasing energy tends to mask problems at the end. Alight gracefully and
complain about sudden 'turbulence' on rollout.

On a powered trike (C150), the friction of 3 wheels and a favorable
configuration of CG and CP will make all of that invisible in light xwinds.
Conversely, a J3 will amplify all of the above and the weeds await the
unwary.