'Is there a contrast between the truth of our lives and the story that we tell of our existence?'

Yes.  Most of us, depending on how we've lived and our individual personalities, will omit or downplay parts of our life in which we feel we acted in a way that was out of character, or casts us in a negative light, when talking about our lives.

Of course there are exceptions to this, on occasion people may prefer to glorify their actions and positively boast about them.  However, they are distorting the 'truth' of their lives, just as much as those who prefer to downplay instances of behaviour that may be frowned upon by society in general.

Often the story that is told will be tailored to its audience.  For example, a conversation with your family on what you did on New Year's Eve might be very different to the conversation you would have with your best friend on the same subject.

It is reasonably common for an individual to feel that they are the focal point of their own story.  Existentialism only formally existed since the 19th Century with Kierkegaard, Heidigger and Sartre, but I suspect that the existentialist view of the world was there long before somebody thought to write about it.

It is important to note that the 'truth' of the story somebody tells about their life will be subjective to a listener who will apply their own values to what they are told.  Even if somebody tells of their life consciously trying not to emphasise or downplay the actions they have taken, a human listener will hear what is said through a filter of their own experiences, knowledge and personal values.