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The updated schematic for the 339 is correct.
It should be okay to use the darlington version of the HOA 1397. It will be more sensitive than a non-darlington. And it may give better operation than the non-darlington.
However it may also need more shielding from stray light. If you find it is too sensitive then it could help if you insert a resistor between the emitter leg of the photodetector and ground.
To get proper operation you'll probably need to do some careful adjusting of resistance values above and below the detector.
It should be okay to use the darlington version of the HOA 1397. It will be more sensitive than a non-darlington. And it may give better operation than the non-darlington.
However it may also need more shielding from stray light. If you find it is too sensitive then it could help if you insert a resistor between the emitter leg of the photodetector and ground.
To get proper operation you'll probably need to do some careful adjusting of resistance values above and below the detector.