All you need for a shapefile is the .dbf, .prj, .shp, and .shx. Importing the shapefile it also looks off, just like in the sample map you sent.
I guessed that the projection might be off, so I used the Info pane to Repair Initial Coordinate System.
In the Standard tab, there are only two projections specifically for Belgium: Belgium Hooijberg and Belgium National Geographic Institute.
I picked Belgium Hooijberg and using that put the Refprv drawing into the right location, as seen in a map with a Bing streets background layer.
Actually, it's close but not exact, which can be seen by using transparent fill color for the areas, so you can see how borders of the areas line up with what are likely defining border lines in Bing. That projection uses a Center latitude of 50.79936222222222 and a Center longitude of 4.367486666666666
Next, I tried Repair using Belgium National Geographic Institute projection. That is way off, exactly as in your original drawing. Looking at the definition of that projection, it uses a Center longitude of 4.367486666666666 but a center latitude of 90, which seems wrong.
Looking into this in greater detail, it appears that the Belge 1972 (EPSG: 31370) coordinate system, as EPSG says, was " Introduced in 2000 as an alternative to CRS code 31300 to avoid the special projection method LCC (2SP Belgium) (code 9803). If software cannot handle latitude of origin 90°N, use latitude of origin = 50°47'57.704"N with Nf = 165372.956 m."
So, translating the degrees/minutes/second origin number into decimal degrees, we get: 50.79936. If we use Repair Initial Coordinate System to assign EPSG 31370, and then we go into the Custom tab and change Center latitude to 50.79936 and False northing to 165372.956, the drawing lines up perfectly.
It turns out there are some rare cases where special projection routines are used for LCC, and those allowed a center latitude of the North pole. Neither 8 nor 9 does those, but I think it should given that Manifold supports EPSG codes that allow a 90 degree center for Lambert Conformal Conic. I've sent in a suggestion to add this.
In the meantime, you can use this projection if you use Repair Initial Coordinate System - More - Custom Tab, and in that Custom tab use 50.79936 for the Center latitude and 165372.956 for the False northing. You can save that as a Favorite (I called my version Belge_Lambert_1972 Alternate) so you can apply it using Repair Initial Coordinate System with one click to any other drawings that use the Belg_Lambert_1972 projection, like the Bedrterr40000 drawing.
What I used for a new Favorite: