This is about assigning a task to ONE person only

[Dante]

Adding a member is not a good solution because when later I run a search with that person name, it will return me tasks in which that person is ALSO a member but not necessarily the person in charge. 
It’s quite strange that a PM tool like TRELLO does have this basic function…..

[Hannah Morgan]

Hi @Dante Welcome to the community! Typically the member field is used as the way to communicate who's assigned to the card, but some people add themselves as members if they are a collaborator or want to be notified of the card. 

I'd suggest making it a standard rule that members should only be used as the assignee, and there should only be one member per card. If someone is a collaborator, then a checklist item within that card can be assigned to them, or if they're just wanting to follow the card, they can simply watch the card to be notified of changes.

An alternative would be to use a separate field to clarify the main owner or person responsible like a dropdown custom field or a label (see Roxanne's post).

[Roxanne Dockter]

I just asked this question too 

https://community.atlassian.com/t5/Trello-questions/Is-there-an-option-to-assign-a-specific-individual-to-a-trello/qaq-p/2002852