I will confess this one left me confused for a while. Yes, the radiance is the same, assuming the emitter is at the same distance (it's not clear from the diagram). The reason for this is because the emitter is always at normal incidence to the detector's view angle.
There's this neat theorem called conservation of radiance which is really helpful in puzzling situations like these, which states that radiance measured at the emitter is the same as radiance measured at the detector, assuming no radiance is lost between emitter and detector. In this case, it becomes obvious the radiance should be the same, as in both cases the emitters are identical up to rotation (which conserves areas, distances, and basically everything you could possibly care about when working with radiance).
There are also some helpful notes under the Wikipedia Talk page for radiance, see the "cosine term" paragraph. The key point is that the cosine term in the definition of radiance does not use the detector's surface normal but the emitter's surface normal. Notably, the sentence "The cosine factor in the denominator reflects the fact that the apparent size of the source goes to zero as your angle of view approaches 90°." highlights this. In your case, the apparent size of the source is quite clearly constant due to the nature of your setup. This can get horribly confusing especially when you see computer graphics papers such as the rendering equation which apparently use the cosine term with the detector's surface normal, but they make entirely different assumptions.
Another, perhaps more intuitive way of deducing this result is by looking at the units of radiance: watts per meter squared per steradian. Also known as power per area per solid angle. Just to be clear, "power" is the power of the emitter, "area" is the surface area of the emitter, and "solid angle" is the solid angle subtended by the emitter from the detector's point of view. Now, for both cases considered:
- power is constant, since the emitter is obviously still emitting the same amount of light in both cases
- area is constant, as the emitter's surface area hasn't changed
- solid angle is constant, because the emitter is held facing the detector at normal incidence in both cases
Therefore, measured radiance is the same in both cases.
That said, I could be wrong on this. Please feel free to correct me if I made a mistake.
EDIT: actually, I don't even know anymore. It's all foggy
EDIT 2: no, I think, radiance should really be interpreted as emitted intensity (power / area) per solid angle (direction). There, that's better still mighty confused though,