Hi,
We cannot control how Adobe classifies the Referrer Type, everyone gets the same logic, and Ads run through a Search Engine are not considered "Search Engine".
That said, you do have control over your Marketing Channels, and you can create logic to group content the way you want using a combination of Referrer, Paid/Natural Search Engine identification, and for this purpose, using your campaign codes on your ads to place them into the channel you want.
You should be aware that by default, Marketing Channels have a 30 day attribution, also Channels like Direct and Internal do not override a more explicitly set channel. You have control over these settings.... For instance, we reduced our Marketing Channels to use 7 Days instead of 30... but I would be wary of changing the behaviour of your Direct / Internal rules as you might start breaking attribution in the same session....
If you are new to Marketing Channels, you should read this first, then I or other people here would be happy to help you with questions about setting up rules:
https://experienceleague.adobe.com/en/docs/analytics/components/marketing-channels/c-getting-started-mchannel
Or, if you really want to ensure you are looking at the Visit, you can set up custom tracking to identify your Referrer Type and track it into an eVar set to Visit level attribution.