It leaves a lot unsaid. Advertising has a lot of connotations and cultural significance. It can evoke strong feelings for one and be a livelihood for another. Simply stating that an advertisement is 'a public notice' doesn't do it justice.
I understand that dictionaries are not encyclopedias, but a little more is warranted IMO.
Example: The Merriam-Webster for example defines "advertisement" as, literally, as "1 : a public notice".
Still,
> but I think the most common understanding of what it is would.
... is probably the reason for their "especially" paragraph and examples, right below the definition.