By the standard security definition of a Pseudo Random Generator, if $G$ is a PRG, then $G'$ such that $G'(0\mathbin\|x)=0\mathbin\|G(x)$ and $G'(1\mathbin\|x)=1\mathbin\|G(x)$ is a PRG. We can build a PRG which output starts by some bits of its input, to some arbitrarily large extent.
This means that feeding a practical source of entropy to a PRG secure by that standard security definition can be very unsafe.
Do we have a stronger security definition of PRG (or some other standard cryptographic construction) that avoids this pitfall, and insure security of the output when the input as some (min)entropy?
Addition: there's nothing wrong with the standard definition of PRG. It is consistent, and useful. It's just not what's needed in some cases, including stretching an imperfect entropy source. I'm asking if we have some stronger security definition of crypto-gismo-with-a-PRG-interface avoiding such issue. Much like for hash we have security in the ROM that's stronger than collision and (first and second) preimage resistance.