Shape length distance wrong in Robinson projection

Quoted from About coordinate systems and map projections

Other projections minimize overall distortion but don't preserve any of the four spatial properties of area, shape, distance, and direction. The Robinson projection, for example, is neither equal area nor conformal but is aesthetically pleasing and useful for general mapping.

You may want to check out this resource:

Useful Map Properties: Distances and Scale

The Define Projection tool only changes the associated projection, it does not modify the geometry of the feature or shapefile. (Useful if you get data with no associated projection. Not useful if you're trying to transform to a different coordinate system.)

Since you ran this before running Project (which mathematically transforms the data into a different coordinate system), your data is still retaining the decimal degree numbers, but with different units. So the geometry calculations still use the same raw numbers that were associated with decimal degrees, but erroneously assume different units (meters).

Try running just Project, and see if the measurements are more what you expect. (If the original world border data has no existing projection, then use Define Projection to assign it WGS84, and then Project to the new coordinate system.)