Just curious, what are the coordinate values that you enter when placing the map in the Georeferencer? This may not be your workflow exactly, but may offer some clues on what's happening in QGIS...
If you're entering Geographic coordinates (eg Lat \ Long in degrees), as in -180, -90, 180, 90
georef_pts.JPG
Then you'll want set the target Spatial Reference to WGS 84. This is to say, it could be difficult to know what coordinate values to enter to georeference a map directly into projected coordinate system, unless you happen to already have some known points on hand... Every projected CS has a different range of values, often very large values, that are not always intuitive (compared to lat, long).
Geored_transform.JPG
Using WGS 84 would georeference your layer with geographic coordinates. And it happens to be displayed in QGIS as equirectangular, since that's the default way to display a layer with a geographic coordinate system
GeoRef_WGS84.JPG
Once it's been georeferenced this way, using the Warp tool (as mentioned above) you can then project the layer as a new layer in a projected coordinate system. Even if you get errors, the tool may have worked. However, (and here's the kicker) if you view that layer in your same map project, maybe it appears just the same
Hey, what gives? It looks just the same? Keep in mind that a QGIS project will apply the current map's existing coordinate system to any new layers added, it will 'project on the fly' to keep all the layers aligned. (Often this is really convenient, except here, when it seems to undo what you're after).
GeoRef_inQGIS.JPG
If you happen to view the projected tif image Outside of QGIS, as an image in your file system, it should look projected (I think as shown in the previous post about using Warp). If you start a blank, new QGIS project and add this as the first layer, it'll appear using the layer's own projected CS.
GdalWarp_export.JPG
As a shortcut, (using this same project on the fly component ). If you have your original georefereneced 'geographic coordinates' layer, and switch your map project's coordinate system,QGIS--> Project --> Properties --> And set the Map Project's Coordinate System to a Projected CS, eg North America Albers Equal Area
QGIS_CRS.JPG
You'd end up with the projected view... It will also probably throw some errors, but this is all pretty general, to you can probably ignore.
prj_on_the_fly.JPG
Anyways, hope some of that may help, in the ever-unfolding saga that is understanding coordinate systems, and their quirks in GIS.