Currently, if an individual is linked to an organization and the organization is marked as primary for that individual, the organization and job title (if any) will display under the person's name at the top of the screen. This occurs even if there is an end date entered for the timeframe of the individual's position in the organization.
It would seem to make sense that an organization / employment would appear in a profile's headline only if it is both primary and current. That is, if there is a past "to" date in the timeframe, the organization / employment would not be displayed below the person's name, even if it is primary.