not believe ghost existing vs not believing ghost exist

Some examples from the web:

• As a mere figment, as a mere ghost existing in your wife's mind... I could hardly be said to have any business.
• We do not believe in detailed common European social legislation.
• Do not believe that more money automatically brings better results.
• You would not believe the stories that I could tell.
• You would not believe how many T-shirts that man has.
• I do not believe that you conspired to kill our father.
• You would not believe the mountains of medical data that the NMRC sent over.
• You would not believe what my moth...
• You might not believe in it.
• I would hope that I would not believe rumors.
• I do not believe you capable of mercy.
• I do not believe that you can kill your own son.
• Whatever Ariadne has done, I do not believe her capable of treason.
• You will not believe where I've just been.
• Ma'am, I do not believe he killed Aaron.
• You will not believe where he's headed.
• But you will not believe what's actually happened.
• You would not believe what people sell on here.
• We do not believe he is stealing your food.
• You would not believe my luck, Al.

Some examples from the web:

• He wants to see if ghosts exist.
• I don't think that ghosts exist.
• I don't believe that ghosts exist.
• The Son and the Holy Ghost exist from all eternity...
• They blatantly deny your ghost exists... and have filed papers with the minister's office... claiming you've overstepped your authority.
• But then, if ghosts exist, it's also possible that aliens exist.
• Perry, not believing in the afterlife is like not believing in astrology.
• you who don't believe that ghosts exist.
• Do you believe now that ghosts exist?
• So all that stuff about not believing in relationships...
• Not believing all the terrible things they say about me committing all those awful crimes.
• I'm sorry for not believing in...
• I was petrified, not believing.
• Clearly she is not believing a word of this.
• Forgive me for not believing you.
• I am not believing what I have just witnessed.
• Talking about not believing in Valentine's day.
• No, your only mistake was not believing in my friendship.
• So, excuse me for not believing his psycho girlfriend.
• I guess it's poetic justice for not believing Sammy.