There are a few different starting methods in use today.
Some of the engines use a starting scheme that requires you to apply a motor on a wand to the shaft nut in order to spin it up while regulating the propane before it will sustain on jet fuel.
Some others require a glow starter battery to be applied to the glow plug during the procedure.
Others are automatic, only requiring the operator to move the trim and throttle lever up in order to start the fully automated starting process.
There are as many starting schemes as you have fingers and toes, and I think that we tend to get lost in the semantics because of the variations, and how we use them in our language.
To me, "onboard start" when used in the same sentence with "full autostart" indicates a difference between two methods.

If you use these same terms independent of each other, they could be construed to mean the same thing.
I guess, in order to answer your question, "Onboard start" is a starting method other than "full autostart". It could differ from full auto in many ways.

Which small engine do you intend to buy?

Harley Condra
BVM REP
JetCat REP