The terms “formal motion” and “procedural motion” mean exactly yhe same thing. They are motions which act upon the processes or predured being followed in the meeting. They are about the running of the meeting.
The problem occurs when people who attend meetings are not familiar with the terminology and they often become confused if the term formal motion is used. The reason is that they belive that all motions are “formal” or “important” or “serious” and so they can be confused.
That is why I always recommend that the term procedural motion is used because its meaning is obvious – it is procedural – the procedures.