Unity's built-in frustum and occlusion culling features work on a per-renderer level. They'll disable the rendering of whole objects that can't be seen, but they don't select individual triangles to cull out of a single mesh.
In both your examples above, the full mesh will be processed by the GPU, because at least part of it is visible.
The extra vertices of the "plane" primitive are unlikely to give any noticeable performance impact unless you're rendering very many of them (modern game engines and GPUs are built to handle character and environment models which are far more complex than this!) so use whichever primitive is most convenient for what you need it to do.
Note that in the two cases you showed, it's really clipping that's at work, not culling. Clipping is the stage in the graphics pipeline that removes triangles (and parts of triangles) that are outside the rendering viewport and the near & far planes. So in both cases, your fragment shader is only being invoked for the pixels of each mesh which are actually in front of the camera. The rest of the surfaces are discarded after some (usually quite cheap) vertex transformations to determine they're outside the rendered area.