If you want to create a Relation between two records, you need to set the ID of the other record.
You want to create a Contact from Organization and want to set the “Organization” Field to the current Organization. You only need to set the field “Organization name” to “$id” (not $accountname). All available ID's are listed on top of the Drop Down List you get on the blue icon next to the text field. (ID of current Record)
This works for all linked fields and also fields inside Projects, Tickets, Events,.. where you could choose between different modules.