Note: These are my experiences so everyone finds their own path!
I use heightmaps in game, but after applying it to the mesh I then add the normal to the vertex as another attribute. I can use this with other attributes so I can ignore areas or apply a bias to a normal, or where I do want to normalise. I pass the normal into the vertex shader along with the rest of the vertex data, the normal will be linear interpolated so I will get normal that accurately is reflected in the pixel shader (just needs to be normalised as lerping denormalises).
This is a far more accurate representation than a normal map, unless you have a high res version of that map than you wont get near the accuracy.
The other piece that I would steer clear of normal maps for terrain. If you ever want to rotate your world, then you need in your pixel shader to sample a point, apply the matrix transform to it then its in world space. If you use normals through the vertex shader, you can apply the same matrix transform once per vertex, not per pixel. Pressure on the pixel shader is the thing I try to avoid. It also means one less bilinear sample in the pixel shader. The transform of the sampling to world space is similar to what you do for calculating normal maps on a model.