There is a simple remedy to address your concerns, which by the way, are valid.
With your wedding invitations have a RSVP card. On the RSVP card, there will be a blank space for the number of people who will or will not attend. When the RSVP card is returned, you will clearly be able to tell if someone invites an uninvited guest by the number of guests filled in on the RSVP card.
If the number of guests does not coincide with whom you invited, then you call up the invitation recipient and let them know that they misunderstood and no extra guests will be accommodated. Do not give reasons or excuses or you will open up the dialogue to defend your choices.
Some brides, with light pencil, write the number of invited guests on back of the RSVP card. So, when they get the RSVP card back, they simply flip it over to see if extra guests were added to the count by the invitee.
Never feel awkward to call people about their error in judgment, just handle it with social grace or have your mother make those phone calls. I am sure you already realize that the way you address the envelope should be clear as to who is invited.
It is a shame that people just automatically think they get to bring someone to a wedding. Perhaps once people mature a bit, they will realize that strangers at a wedding is very gauche. A wedding is an intimate affair and if you do not know someones name, they don't belong there.