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“I'm not sure which is the "observer"” [excerpt]

The one you're standing on while looking at the other.

“and which is the "observed", ” [excerpt]

The one that you're not on, but you are looking at.

“but it seems like the displacement is relative to whichever one is standing on the Earth, and that would be the same to them regardless of which scenario is causing the displacement.” [excerpt]

I'm not sure I follow.

If there is any motion, there will be displacement.

If what you're looking at is moving, its Light-time correction.

If you're moving, then its the Aberration of light.

(The way I understand it, if you and what you are looking at are both moving the same direction and speed, the Aberration of light and Light-time correction cancel each other out, and what you see is what you get)

“What is the calculated difference in observed displacement between the two scenarios?” [#1135]

~2.1° (Light-time correction) versus ~0.00583° (Aberration of light).

“How do you calculate light-time correction in the case of a two-body geocentric (orbiting) model. The distance between the two bodies is constant.” [#1136]

I don't have a formula handy, but its pretty simple.

You take the time it takes for the light to reach the Earth from the orbiting Sun, and then calculate how many degrees across the sky the sun travels in that time.

Off the top of my head:

 transit time in seconds = distance in meters ÷ speed of light in meters per second
 displacement in degrees = earth rotation speed in degrees per second × transit time in seconds

I think thats how it works. (Not sure, I haven't double checked it)