Is it better to bootstrap a step-down converter from the input or the output?
Pick whichever has the most convenient voltage. Usually the supply providing power to the output is far too high voltage and the logic supply is closer.
But in your second drawing you imply you want to grab the output of the stepdown converter. That doesn't make sense since it can never start up. You would need to grab the output of the supply you are stepping down.
As for the third drawing, I don't understand how it is possible to use Vout if Vout is truly the output of the ACT4088.
CLARIFICATION:
You have three &quot;possible&quot; options: Vcc, Vin, and Vout. Your third diagram us using Vout. But if Vout is zero before starting up, and needs to not be zero to run then it can never start up. Catch-22. You can't use Vout. You can, however, use Vin but it is often too high.

From Linear Technology LTC444 Datasheet