The knife tool is one of the things I use most in blender. I use it to add vertices with out adding a Loop/cut/slide. I recently got a new laptop and installed blender on it, and it creates a random face when using the cut tool.
Usually this is what I do to adda vertex, as you can see in the picture.
But as you can see instead of only adding a vertex it connects to the corner of the object.
Alt
+D
with vertice selected where you'd like to add another one (you may want to slide added vertex for accurate positioning). I have to admit it isn't the best workflow as it doesn't create clean topology which you can handle with easily. $\endgroup$