Hydrocracked Group III bases were considered and sold as full synthetic around the world way before they were called that here in the US. Both by Mobil and Castrol. Non-hydrocracked bases are not considered synthetic here nor elsewhere.
The only place they aren't sold that way is in Germany. Everywhere else they are.
It's also because the Group designations expressly exclude method of manufacture. They are performance requirements. Hydrocracking (which defines synthesis in this discussion) isn't in and of itself required to meet the performance requirements of any Group I-III designation.
Conversely you could have a hydrocracked, synthetic base that is Group I or II. That process is progressive and cumulative, not a one-and-done.