Hello, Is there a way to limit the views available to users in a single field's quick and advanced finds? I'm trying to limit the views available for the field 'Assignment->To External Organization' without limiting those available in other organization fields.
Quick view is using the default view (per account) for the given item. In Advanced find the views that are listed are the views that user's role has access to.
So you cannot do what you want. However you can create a view named eg External Orgs and when your users try to assign an item to external org they can use this view through advanced find to filter out the entries. However this functionality requires SP16 or newer.
Each user can define what view is the default view for his role. So even if you create a view you won't have guarantee that this will be user's default view unless it's the only view that user has access.