Get off the mark also means IMO
to startand is of course only
related to
quick off the mark
(which indicates quickness in starting)
It means to start because you're leaving the starting blocks/marks in a running race.
The site I mentioned shows:
quick off the mark
which I see as
quick (to get) off the mark
-----
--Fast to start or try something
----
thus
(to get) off the mark IMO is clealy
to start
Another related idiom:
Get off the dime
is mentioned by Spears as meaning
to
start moving, to get out of a stopped position.