>Not lying about one thing does not Change the fact that we should not trust them

That's... not a very useful way of looking at things. Whether or not you trust what someone says should be based on a number of factors, their trustworthiness, sure, but also their motivations and what they would gain by telling the truth or lying.

Coca Cola would probably have a lot to gain by pushing the idea they invented the image of Santa Claus. The fact that it would so easily disprovable if they were to claim that they did is also another reason to expect them to be telling the truth in this instance.

That said they do seem to push the idea that they somehow solidified his appearance from a number of other depictions, which isn't really true, or at least they're vague enough in their wording to imply it without stating it. The modern image of Santa existed before Coca Cola began advertising.

https://www.snopes.com/fact-check/the-claus-that-refreshes/

>There was a period of overlap during which the modern Santa Claus character coexisted with other Christmas figures and other versions of himself, as his now-standard appearance and persona jelled and his character grew in popularity to become the dominant (secular) Christmas figure in the western world. However, that period had ended before Coca-Cola began utilizing Santa for their holiday season advertisements. As noted in a New York Times article published in 1927, four years before the appearance of Sundblom's first Santa-based Coca-Cola ad, the Santa Claus figure rendered by Sundblom was based upon what had already become the standard image of Santa:
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>Coke's annual advertisements featuring Sundblom-drawn Santas holding bottles of Coca-Cola, drinking Coca-Cola, receiving Coca-Cola as gifts, and enjoying Coca-Cola became a perennial Christmastime feature which helped spur Coca-Cola sales throughout the winter (and produced the bonus effect of appealing quite strongly to children, an important segment of the soft drink market). One might therefore fairly grant Coca-Cola some credit for cementing the modern image of Santa Claus in the public consciousness, as in an era before the advent of television, before color motion pictures became common, and before the widespread use of color in newspapers, Coca-Cola's magazine advertisements, billboards, and point-of-sale store displays were for many Americans their primary exposure to the modern Santa Claus image. But at best what Coca-Cola popularized was an image they borrowed, not one they created.