There have been two generations of Star liveries, the first where the entire livery is placed on a section of the plane, as shown in PremEx's first Thai photo. The second just has the logos, as shown in PremEx's second linked photo.

The operating carrier has their logo or livery first. Otherwise it is a normal airplane.

I believe one airline (Lauda?) has painted the livery on a tiny turboprop!! so the logos are REALLY small!