I think you are correct. Your Trigger is on an update in the DocCloud Projects table. When that update starts the trigger, the Automation gets the Record ID of the record that updated in DocCloud Projects table.
In your Action, you are telling Airtable to update an individual record by providing a Record ID, the only Record ID you have so far is the record in DocCloud Projects table.
You want to tell Airtable which record in Projects & Tasks to update.
You will need to add a Find Action to locate the correct record in the other table.
However, just looking at your screenshot, is it safe to assume that records in Projects & Tasks are linked to DocCloud Projects? Are you wanting to reflect the Kanban Status of an updated record in DocCloud Projects in the Projects & Tasks table. You should be able to add a Lookup type field to reflect the state of that record without any Automation.