However when I export it to version FBX6.1 ASCII (available in Blender 2.79) then I do get the animation.
Now it is probably not the intended method, because with the ASCII export my mesh doesn't get imported to the model editor. So you'll first need to export your meshes as above. When saving the editor will crash but the .mdl is saved. Then export again in ASCII for the animations and when you save that the editor creates the .ani file.