This one:
"the current supplied remain constant and the batteries just drain less"
The LED current will be unaffected by the addition of the second identical parallel battery.
V = I x R
In this circuit you are doubling the battery, but not changing the output voltage (two identical 9V batteries in parallel is still a 9V output).
On the load side, the resistor and LED have not changed (that's the R in Ohm's law). Please note an LED is not accurately modeled as a pure resistance, but a complete explanation of that is not necessary to understand the answer to your question.
No change in V; No change in R; ...therefore NO CHANGE in I (current)
E = V x I x t
What does change is the total potential energy in this circuit. If you double the battery count, the total current sourced to the LED will be unchanged, but the current supplied by each battery will be 1/2 of the total. Because the batteries are supplying half the current as before, they will last twice as long.
Energy is voltage times current times the time the current is supplied at that voltage. A 1000mAh Alkaline battery means that it can supply 1A at ~1.4V for ~1 hour.
So...
No change in E; No change in V; ...therefore battery life (time) is INVERSELY proportional to current
The circuit you refer to Data Sheet (page 23, is merely an output signal to a user that an error has been detected. The detection circuitry is another matter.
The manufacturer has spent a lot of time and money in developing their detection circuit.
"How would such circuit be implemented to be inexpensive and reliable?"
Achieving Reliability is a broad subject involving both mechanical and electrical design.
The manufacturer states, "The error output of SKiiP®
3 V3 is short circuit proof.". Using an open collector means you can short the open collector to Ground with no ill effects.
Best Answer
Yes, this is exactly the intended purpose. It's called a wired-OR or wired-AND, depending on your logic. (Or, as Tony points out, even a wired-NOR!)
However - you should not do this if you just want twice the amount of current sinking capacity. Only if you want the feature of being able to switch using separate inputs. Current will likely not be shared equally between the devices.