Select the plane as the object to be split and the circle as the cutter.
The user has no choice in the direction the circle is projected
You could extrude the circle and create a cylinder and then use the edge of the cylinder as the cutter. Notice that trimming with the edge of the cylinder produces the same result as trimming with the original circle.
Also notice that when trimming with curves you have no option to ask Rhino to only produce a result if the curve is within tolerance. That’s one advantage in trimming with a surface. Rhino will only use an edge as a cutter if it is well within tolerance of the surface being cut.
You want a method that will trim using the edge of a surface that works only when the trim produces a result so both surfaces can be joined. The way to do that is to use the surface as a cutter. In your example it will fail, but if you were to make the surface so that the edge is within tolerance the trim would succeed. And then the edges would join.
The problem is that method has bugs that make it not as reliable as it could be. Those bugs had been fixed years ago but then have reappeared.
Bobi posted an example of an edge that should trim but doesn’t. The trimming does work correctly in Rhino2 and Rhino4.
IMO, Project should be never used when the cutter object is the edge of a surface when the surface that edge belongs to has been selected as the cutter. As far as I know there is no possible circumstance where projecting any cutter surface edge is going to do anything but make the trimming less accurate.
Yes I know and you don’t have much other choice, but Split or trim should not fail as often as it does.
There are many cases (like your example) where the edge of a surface is well within tolerance and yet Split or Trim fails.