You are not serving yourself well as you seem to have a strange relationship with the truth.

If what you say is true that you made an honest mistake and you have a good reference from your direct supervisor it would be ill-advised not to be truthful.

If asked by the potential employer, just explain the situation.

Take it easy. Don't damage yourself to the degree when your future productivity is affected.

If someone came to me with that story, the vetting by Google would heavily outweigh the story.