Yes! These terms are interchangeable and refer to the same thing. This also goes for conversion disorder, central sensitization, overactive sympathetic nervous system, functional neurological disorder, etc.
Not really. Somatization/conversion disorder are recognised by the medical community whilst TMS is at best viewed with scepticism. I think the crucial thing is the Freudian psychological theory of causality that Sarno saw as a defining factor of TMS. Many of Sarno’s TMS theories were actually nothing new or groundbreaking but adding the Freudian angle made it fundamentally different.