How many types of speech impediments are there?

Many! Speech impediments are now called "fluency disorders." There are many besides the typical stutter that comes to mind most often. See this discussion: https://www.urmc.rochester.edu/encyclopedia/content.aspx?contenttypeid=134&contentid=237 Many famous people have had fluency disorders and learned to overcome them. For example see these folks: http://www.stutteringhelp.org/famouspeople.