Well, it is not exactly same, but it will produce exactly the same answer (as long as that n is a unit normal). We are using different parts of the same triangle and different relationships, to arrive to the same effective result (might differ by a sign but that isn’t important). In your method you are projecting (p-q) onto the unit normal. In my method, I was projecting p onto the unit normal, and subtracting the displacement of the plane. It results in the same value.
Never thought about the fact that I was actually projecting…