Cfrtjryk

For example, my son's birthday is Christmas Eve. When I tell people this I generally add how I have to tell family and friends not to go cheap and buy him a birthday present but say it's also for Christmas. In fact, I have never had to do this. I only say this because so many people have brought this up I preempt them with this "anecdote".

"That story I told wasn't exactly true, it was a/an ______"

You may be indulging in polite fiction.

From Wikipedia:

A polite fiction is a social scenario in which all participants are aware of a truth, but pretend to believe in some alternative version of events to avoid conflict or embarrassment.

Note that "all participents" doesn't necessarily include everyone the events are told to; as in your case and this example offered in the article:

... a couple that has had an argument, after which one of them absents him or herself from a subsequent social gathering, with the other claiming that he or she is "ill".

First of all, I'd say that most people know for sure if a story they tell is true or not. (If they honestly don't know, then it falls into the realm of the personal equivalent of an urban legend, a fable, or simply a speculative memory.)

Your story is technically a lie (it's not truthful—or at least not entirely truthful), but it's being said in order to enhance something for dramatic effect rather than to mislead per se.

In that sense, it's more like an embellishment.

From the Merriam-Webster thesaurus entry:

2 the representation of something in terms that go beyond the facts

// the actorʼs penchant for embellishment suggests that his memoirs would be more appropriately shelved in the fiction section

Synonyms of embellishment

caricature, coloring, elaboration, embroidering, embroidery, exaggeration, hyperbole, magnification, overstatement, padding, stretching

Words Related to embellishment

amplification, enhancement

fabrication, misrepresentation

fudging, hedging

hype, puffery

superlative

I’ve heard the word apocryphal used recently in similar situations. For example, a story about the actions of a famous celebrity which is particularly consistent with their perceived character, with no firm evidence or sources to back it up, but it is a nice story nonetheless.

From Collins

1. of questionable authenticity


2. (Bible) (sometimes capital) of or like the Apocrypha


3. untrue; counterfeit

In the context I’ve heard it used, it almost certainly implies a (well intentioned) lie that is also a good story.