Just to add to what Robin has said and answer your last question.
"..are the underlying concepts related if you consider the Pali origins of the words i.e. do they have similar roots in the language?"
Samadhi (Sanskrit: समाधि, ...), also called samāpatti, in Buddhism, ... is meditative absorption, attained by the practice of Jhana. In samadhi the mind becomes still, one-pointed or concentrated while individual awareness remains present.
Sometimes upacara samadhi and appana samadhi is used to distinguished between threshold concentration and absorbed concentration.
Samatha (Pāli), (Sanskrit: शमथ, śamatha[note 1] is the Buddhist practice (bhavana) of the calming of the mind (citta) and its 'formations' (sankhara). This is done by practicing single-pointed meditation most commonly through mindfulness of breathing. Samatha is common to all Buddhist traditions.
Paring it down to a few words. Samadhi is concentration and Samatha is a practice.
As to your understanding:
My understanding is that samatha seems like the process of integration and samadhi seems like the result of integration.
Very close but if you replace "integration" with "concentrating" spot on.